Habilitation a Diriger des Recherches

Habilitation a Diriger des Recherches
UNIVERSITE DE PROVENCE
Habilitation a Diriger des Recherches
presentee par
Jean-Marc VIREY
specialite :
Physique Theorique : Particules et Cosmologie
RECHERCHE DE SIGNAUX DE NOUVELLE PHYSIQUE
EN PHYSIQUE DES PARTICULES ET EN COSMOLOGIE
Soutenance le 14 Decembre 2007, devant le Jury compose de :
Mme Anne
EALET
M. Philippe AMRAM (Rapporteur)
M. Alain
BLANCHARD
M. Jer^ome MARTIN (Raporteur)
M. Patrick PETER
M. Jean-Marc RICHARD (Rapporteur)
M. Thomas SCHUCKER
M. Pierre
TAXIL
.
2
Cette Habilitation a Diriger des Recherches, ainsi que la plupart des travaux presentes,
ont ete realises au sein du Centre de Physique Theorique de Marseille et de l'Universite
de Provence. Je tiens a remercier toutes les personnes qui participent a l'existence de ces
structures : enseignants-chercheurs, chercheurs, personnels administratifs et doctorants.
Qu'elles soient ici profondement remerciees de leur disponibilite a mon egard.
A ma Famille et a mes Amis, qui m'ont toujours soutenu au cours de ces annees de
travail, et en particulier au moment de la redaction ou ils ont accepte mes changements de
comportement et d'humeur.
Ma reconnaissance va a l'ensemble de mes collaborateurs scientiques sans qui les
travaux presentes n'auraient sans doute jamais vu le jour. En particulier, j'adresse un
grand merci aux membres du groupe Renoir du CPPM (Anne Ealet, Charling Tao, Andre
Tilquin, Dominique Fouchez et Alain Bonissent), aux membres du groupe Cosmologie du
LAM et plus precisemment a Alain Mazure et Jean-Paul Kneib qui ont permis la mise
en place d'une activite interdisciplinaire entre les trois laboratoires CPT-CPPM-LAM. Je
salue chaleureusement les eorts de Diane Talon et de Sebastian Linden, doctorants avec
qui j'ai le plaisir de travailler actuellement.
Je remercie profondement Anne Ealet, Alain Blanchard, Patrick Peter et Thomas
Schucker qui, malgre leurs emplois du temps extr^emement bien fournis, ont accepte d'^etre
dans mon Jury.
Mes remerciements vont encore a Jean-Marc Richard, Jer^ome Martin et a Philippe
Amram, qui ont accepte d'^etre mes rapporteurs et ceci malgre le court delai dont ils ont
dispose et l'importante quantite de travail que cela a represente.
Merci a Jacques Soer, qui s'est revele ^etre un collaborateur inestimable par sa grande
competence, sa gentillesse et sa disponibilite. Je lui dois, en grande partie, les quelques
notions que je possede de la Physique du Spin et je regrette qu'il n'ait pas pu faire partie
du Jury.
Merci a Christian Marinoni pour nos nombreux echanges scientiques qui ont gones
mon enthousiasme pour la cosmologie. J'espere que cette collaboration naissante se poursuivra encore tres longtemps.
Les mots sont insusants pour remercier Pierre Taxil, qui a su non-seulement me
supporter mais aussi m'encourager tout au long de ces annees de travail. Sans Pierre, ma
vie serait autre et je ne serais pas la ou j'en suis aujourd'hui.
3
.
4
Contents
Introduction
7
1 Au-dela du modele standard de physique des particules : recherche de
nouvelles particules et de nouvelles interactions
11
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Nouvelle Physique aupres de RHIC-Spin . . . . . . . . . . . . . . . . . . .
1.2.1 L'experience RHIC-Spin et la crise du spin . . . . . . . . . . . . . .
1.2.2 Recherche de Z 0 leptophobes . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Interaction de contact et nouveaux bosons pour RHIC-Spin ameliore
1.2.4 Contraintes sur le secteur scalaire avec des neutrons . . . . . . . . .
1.2.5 Discussion et prospective . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Recherche de la supersymetrie violant la R parite au LHC . . . . . . . . .
1.4 Nouvelle Physique aupres de HERA polarise . . . . . . . . . . . . . . . . .
11
17
17
22
36
46
67
69
81
2 Determination des parametres cosmologiques et proprietes de l'energie
noire
109
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Le Modele Cosmologique Standard . . . . . . . . . . . . . . . . . .
2.1.2 Modeles d'energie noire . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 La degenerescence geometrique . . . . . . . . . . . . . . . . . . . .
2.2 Supernovae : biais et prospectives . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Biais d^u a une equation d'etat dynamique . . . . . . . . . . . . . .
2.2.2 Biais d^u a M . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Biais d^u a une evolution non-lineaire de q(z) . . . . . . . . . . . . .
2.2.4 Prospective SNIa : champ large ou profond ? . . . . . . . . . . . . .
2.3 Analyses combinees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Combinaison SN+CMB et prospective avec WL sur les proprietes de
l'energie noire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Contraintes sur la courbure et une energie noire dynamique simultanement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Test cosmologique a partir de la cinematique des galaxies . . . . . . . . . .
110
110
118
126
130
131
155
166
176
192
192
207
217
Bibliographie
245
5
.
6
Introduction
La recherche d'une unite cachee derriere l'immense diversite de la Nature a toujours passionne les physiciens. Aujourd'hui nous avancons de plus en plus vers une comprehension
uniee des lois de la physique gouvernant a la fois les grandes structures de l'Univers,
comme les etoiles, les galaxies et les amas, et la structure intime de la matiere, comme les
atomes, les noyaux et les nucleons.
La precision experimentale obtenue par les nombreuses experiences en physique des
particules a permis de valider le Modele Standard des interactions fortes et electro-faibles
decrit dans le cadre de la theorie quantique des champs. Cependant de nombreuses questions restent encore ouvertes et on attend avec impatience les premiers resultats du LHC,
d'ici deux a trois ans, pour conna^tre quelles manifestations de nouvelle physique permettent de decrire la voie que la Nature a choisie. A titre d'exemple de ce jeu n entre experiences, modeles et theories, a la base de toute demarche scientique, citons les
predictions du modele standard electro-faible concernant les proprietes des bosons W et
Z 0 . Celles-ci sont veriees par les experiences jusqu'au niveau des corrections radiatives,
mais elles necessitent l'introduction du boson de Higgs, une particule "scalaire" fondamentale, qu'il reste encore a decouvrir. D'un point de vue theorique ce "modele" pose de
nombreux problemes car la theorie quantique des champs fournit en general des predictions
divergentes en presence d'un champ scalaire fondamental (problemes hierarchique et de
non-naturalite). On introduit alors des modeles ou theories allant au-dela du modele standard permettant de s'aranchir des dicultes liees a la presence de ce champ scalaire. Les
deux grandes possibilites sont l'existence soit d'une sous-structure (pour le Higgs et/ou
les fermions fondamentaux (leptons, quarks)) soit d'une nouvelle symetrie liant bosons et
fermions, appelee "supersymetrie" et qui peut intervenir dans le cadre plus general de la
supergravite ou des theories des cordes. Avec ces dernieres theories on entre dans une
troisieme voie qui tente d'elargir le cadre de la theorie des champs an de pouvoir decrire
l'interaction gravitationnelle. Les theories des cordes, ainsi que ses concurrents comme la
geometrie non commutative ou la gravite quantique a boucles, etablissent alors un lien
naturel entre le monde microscopique et les plus grandes echelles de l'Univers.
Il est passionant de s'interesser a ces questions particulierement theoriques mais qui
semblent plus mathematiques que physiques. Il est excitant de realiser que les resultats du
LHC ainsi que la grande precision des nombreuses observations qui vont ^etre entreprises
dans la prochaine decennie dans le domaine de la cosmologie vont permettre de tester certains modeles derives de ces dierentes theories.
7
En eet, l'etude globale des caracteristiques de l'Univers a enregistre des progres impressionnants depuis ces vingts dernieres annees et les projets observationnels en cours de
discussions laissent entrevoir des perspectives tres allechantes. La cosmologie moderne repose sur trois piliers : la comprehension de l'expansion, la nucleosynthese primordiale ainsi
que la prediction et l'observation du rayonnement cosmologique micro-onde. Le cadre est
celui de la Relativite Generale et des lois de la physique microscopique et macroscopique.
Ces diverses lois sont veriees a un tres haut degre de precision localement, mais la cosmologie extrapole leurs applications a des distances tres grandes et a des instants tres
lointains. Le Modele Standard cosmologique repose sur de solides bases theoriques mais
necessite l'introduction de nouvelles composantes pour ^etre en accord avec les diverses observations. Plus precisement, dans le modele dit de "concordance", la matiere ordinaire ne
represente qu'environ 5% du contenu total de l'Univers qui semble posseder une geometrie
plate. La matiere noire entre dans une proportion de 25% et joue un role cle dans la dynamique des galaxies et des amas ainsi que dans les modeles de croissance des structures a
partir des uctuations primordiales de densite. L'energie noire represente 70% du contenu
energetique et rend compte de l'acceleration de l'expansion.
Les contraintes sur la matiere noire indiquent que le uide associe est sans collisions
avec la matiere ordinaire et donc que l'interaction serait purement gravitationnelle. Les
modeles de microphysique, comme la supersymetrie, fournissent des candidats naturels
que l'on cherche a detecter par des methodes tres diverses allant de la physique des collisionneurs a celles des astroparticules via l'etude des rayonnements cosmiques. Une autre
possibilite, radicalement dierente mais de plus en plus critiquee dans la communaute,
serait de changer les lois de la dynamique gravitationnelle a l'echelle des galaxies et des
amas. On espere que les futures donnees experimentales et observationnelles permettront
de separer ces deux classes d'interpretation.
Concernant l'energie noire, la situation est bien plus dramatique (ou interessante !)
car, d'une part, cela concerne la composante aujourd'hui dominante, et d'autre part, que
de tres nombreuses et diverses explications sont possibles. Les astronomes et les "relativistes" vont preferer une constante cosmologique qui peut ^etre introduite naturellement
dans les equations d'Einstein mais qui soure alors du probleme de "concidence" (pourquoi
matiere et energie noire sont elles du m^eme ordre de grandeur aujourd'hui bien que leurs
dynamiques soient tres dierentes ?). Les "physiciens" vont alors proposer une origine
microphysique a cette constante cosmologique en l'associant a l'energie du vide. Ils sont
alors confrontes au probleme de son estimation qui est divergente en theorie quantique
des champs : c'est le "probleme de la constante cosmologique". An de resoudre ces
problemes, deux scenarios distincts sont envisages: soit il faut eectivement rajouter une
nouvelle composante, soit les equations de la dynamique sont a changer. Dans le premier
cas et si on veut resoudre (partiellement) le probleme de concidence, on introduit alors un
champ scalaire (e.g. les modeles de quintessence). Dans le second cas on modie ou on
etend le cadre de la relativite generale (ou des seules equations de Friedmann).
Un autre aspect tres incertain de la cosmologie standard concerne la periode d'ination
dans l'Univers primordial qui rendrait compte du probleme de causalite mis en evidence
par la tres grande uniformite du fond dius cosmologique, ainsi que de l'apparente plati8
tude spatiale de l'Univers. A nouveau les modeles d'ination necessitent soit l'introduction
d'un champ scalaire soit une modication des equations de la dynamique.
On peut donc noter un certaine similitude entre ces trois grandes enigmes du modele
cosmologique standard que sont la matiere noire, l'energie noire et l'ination. Il faut
cependant remarquer que les domaines d'application de ces trois composantes sont extr^emement dierents dans l'espace et dans le temps. Neanmoins, les physiciens theoriciens
imaginent de nombreuses connexions possibles amenant a une tres grande diversites des
modeles. Observationellement certaines quantites mesurees ne dependent que d'une (ou
deux) de ces composantes, mais il est interessant de realiser que les modeles de formation
de structures sont sensibles a l'ensemble de ces inconnues dont une meilleure connaissance
est indispensable pour obtenir une vision plus complete et une meilleure comprehension.
Les nombreuses donnees de sources dierentes et de tres bonne qualite vont ainsi permettrent de relier des problemes aux frontieres de la physique theorique, de la physique des
particules, des astroparticules, de la cosmologie et de l'astrophysique, en esperant qu'une
vision plus uniee des diverses interactions fondamentales et des dierentes echelles de
l'Univers en resulte.
Dans cette habilitation je vais aborder certains de ces problemes en distinguant les
travaux realises dans le domaine de la physique des particules de ceux eectues en cosmologie. J'ai entrepris des etudes de certains defauts de chaque "modele standard" a travers des
analyses phenomenologiques des diverses manifestations des modeles de nouvelle physique,
en insistant particulierement sur les eets experimentaux et/ou observationnels.
La premiere partie concerne la recherche de signaux non-standards aupres de collisionneurs. Certaines de ces etudes font suite a mes travaux de these qui se concentraient sur
la manifestation de nouvelles particules ou de nouvelles interactions lors de collisions avec
faisceaux polarises. Nous verrons qu'une nouvelle interaction purement hadronique (sousstructure ou nouveaux bosons de jauge), faible devant QCD, peut rester cachee et se manifester uniquement au travers d'observables decrivant les eets de polarisation (asymetries
de spin). Nous etudierons aussi les informations cruciales et uniques que fournissent ces
asymetries de spin sur la structure chirale et la structure scalaire des nouvelles interactions.
Les modeles de nouvelles physiques concernes sont les manifestations "a basse energie" relativement generique des modeles de sous-structures et de grande unication, ainsi que
certains modeles plus speciques en supersymetrie ou derives des theories des cordes. En
particulier, un article discutera les manifestations de la supersymetrie si la R-parite est
brisee, dans le canal de production d'un seul quark top au LHC.
Dans la seconde partie je presente mes travaux en cosmologie, qui concernent essentiellement la determination des parametres cosmologiques avec un accent particulier sur
les proprietes de l'energie noire. Les premiers s'interessent au probleme des biais d^us aux
hypotheses realisees dans le processus d'interpretation des donnees. Nous verrons que des
hypotheses sur les proprietes de l'energie noire ont un fort impact sur la determination
des autres parametres, et inversement. Les seconds concernent directement l'extraction de
contraintes sur l'energie noire a partir des donnees observationnelles combinant plusieurs
9
sondes cosmologiques. Un travail de prospective realise dans l'optique de l'optimisation
de futurs projets est egalement presente. La derniere partie propose un nouveau test de
la cosmologie base sur l'utilisation des proprietes cinematiques des galaxies (vitesses de
rotation) an de construire des "chandelles" et des "regles" standards.
10
Chapitre 1 : Au-dela du modele
standard de physique des particules :
recherche de nouvelles particules et
de nouvelles interactions
1.1 Introduction
Nous allons voir rapidement, d'une part, les raisons qui motivent l'existence de modeles de
nouvelle physique allant au-dela du modele standard a travers les limitations de ce dernier,
et d'autre part, l'inter^et d'utiliser divers collisionneurs et en particulier ceux avec faisceaux
polarises pour tester ces m^emes modeles. Ces deux aspects ont ete largement discutes dans
ma these et ne seront donc que succintement rappeles dans cette habilitation ou j'insisterai
surtout sur les devellopements recents.
Le Modele Standard, base sur le groupe de jauge SU(3)c x SU(2)L x U(1)Y et brise
spontanement versple groupe SU(3)c x U(1)em via le mecanisme de Higgs a une echelle
d'energie vEW = ( 2GF ) 1=2 246 GeV , decrit de facon extr^emement precise les mesures
experimentales realisees jusqu'a present (a quelques exceptions pres). Cependant, il est
communement admis que ce Modele Standard n'est en fait qu'une theorie eective a basse
energie d'une theorie plus etendue qui permettrait de resoudre de facon naturelle un grand
nombre de problemes inherents au Modele Standard dont une liste non exhaustive peut
^etre la suivante :
secteur scalaire du Modele Standard : nature du Higgs et origine de la brisure
electrofaible,
origine de la hierarchie des masses des fermions (e.g. me;u;d ' 10 5vEW mais mt vEW ),
grand nombre de parametres laisses libres par la theorie et restant a xer par l'experience,
origine de la violation de la parite des interactions faibles,
origine des trois generations et donc de l'apparente duplication des quarks et leptons,
11
quantication de la charge electrique,
origine du connement des quarks et du spin des nucleons.
A ces dicultes nous pouvons ajouter le desir, pour un grand nombre de physiciens,
d'unier les dierentes interactions et/ou d'unier les particules elementaires (leptons et/ou
quarks et/ou bosons). Avant de detailler les diverses pistes envisagees, insistons sur le premier point de la liste qui traite du probleme de l'existence d'un champ scalaire massif en
theorie quantique des champs.
Le potentiel du champ de Higgs H (doublet complexe) est suppose ^etre donne par la
relation suivante : V (H ) = m2H yH + (H yH )2. La symetrie SU(2)L x U(1)Y est brisee
spontanement vers U(1)em lorsque m2 < 0 et que lepchamp de Higgs acquiert une valeur
moyenne dans le vide non nulle (h0jH j0i = (0; v= 2)T ). Concretement, sur les quatre
degres de liberte initiaux, trois sont absorbes par les bosons W et Z 0 an de leur fournir
une masse, le dernier correspond au boson de Higgs, qui a un spin 0 (scalaire) et une masse
:
p 2 p
mH =
2m = 2 vEW :
(1.1)
Le fond du probleme (dit de \naturalite") vient du comportement a haute energie de cette
masse, ou plus precisemment du couplage quadratique . Denissons le cut-o de notre
theorie par , qui dans le meilleur des cas (ou le pire !) peut ^etre associe a l'echelle de
Planck ( = MP 1019 GeV ) et qui represente l'echelle d'energie a partir de laquelle
le modele standard cesse d'^etre une bonne representation eective de la realite. Gr^ace
aux corrections radiatives (controlees par les equations du groupe de renormalisation du
modele standard) qui font evoluer avec l'energie les valeurs de l'autocouplage , on peut
relier cette echelle de nouvelle physique a la masse du Higgs mH . La volonte d'avoir le
modele standard valide jusqu'a l'echelle implique plusieurs criteres comme la stabilite
du vide et la perturbativite du modele [1, 2].
La stabilite du vide du modele jusqu'a une echelle d'energie Q, requiert (Q) > 0 pour
Q < et fournit une borne inferieure sur mH qui est d'autant plus elevee que est grand.
Le critere dit de \perturbativite" du modele standard correspond a la volonte de maintenir
le couplage quadratique dans un regime perturbatif : (Q) < 4 pour Q < . On obtient
ainsi une borne superieure sur mH qui est d'autant plus basse que est grand. La gure
1.1 donne les zones permise et interdites pour les valeurs de mH en fonction de l'echelle de
nouvelle physique , tiree de [2].
On en deduit donc que la decouverte d'un Higgs lourd indique une echelle de nouvelle
physique tres proche. Un Higgs leger nous donne le moins d'information possible. Seul
un higgs (leger) de masse comprise entre 150 et 190 GeV permet au modele standard
d'^etre parfaitement deni jusqu'a l'echelle de Planck. Une decouverte dans cette gamme
d'energie serait le resultat le plus navrant car aucune piste vers la nouvelle physique ne
nous serait fournie avec l'indication supplementaire qu'elle n'est pas necessaire avant 15
ordres de grandeur en energie ! En revanche, l'absence de decouverte du Higgs devrait
12
Figure 1.1: Limites theoriques sur la masse du Higgs, tirees de [2]
sonner le glas du modele standard, ou au moins du mecanisme de Higgs ...
Dans le cas ou = MP , il n'est pas necessaire d'introduire de nouvelle physique
(jusqu'a l'echelle de Planck ou forcement les eets de gravite quantique doivent commencer a se faire sentir) mais on est alors confronte au probleme hierarchique. En eet,
le modele possede alors un grand desert entre l'echelle electrofaible et la masse de Planck.
D'autre part, les corrections radiatives a mH (ou ) necessitent un ajustement tres n a
l'epoque de Planck an d'obtenir mH vEW . On peut comprendre schematiquement
le probleme de la facon suivante [1]. Les corrections radiatives a la masse d'un boson (scalaire) sont quadratiquement divergentes. A l'ordre d'une boucle on a la relation
m2H (Q = vEW ) = m2H (Q = ) + a2 ou a est une constante. Avec = MP le rapport
mH (Q = )= doit ^etre xe a 17 decimales pres ![1] Ce probleme d'ajustement tres n, ou
probleme hierarchique, nous fait penser que le modele du grand desert jusqu'a l'echelle de
Planck n'est pas un scenario realiste. On s'attend donc tres fortement a ce que le probleme
de l'existence d'un champ scalaire soit synonyme de l'existence d'une nouvelle physique a
une certaine echelle (clairement indeterminee et non forcement a l'echelle du T eV ...).
Du point de vue experimental les contraintes sur la masse du Higgs sont de nature
directe ou indirecte. Tant que le LEP fonctionnait au pic du Z la recherche du Higgs
13
etait negative. Cependant, le Higgs inuence les corrections radiatives. La contribution
est de nature logarithmique, ainsi les contraintes sont assez faibles et sont contaminees
par les incertitudes sur la masse du quark top dont les corrections sont quadratiques. Les
mesures de haute precision au LEP sur de nombreux observables dierents permettent de
contraindres les eets des corrections radiatives et ainsi d'obtenir des contraintes sur mH .
Les resultats
indirects obtenus sont donnes sur la gure 1.2 et indiquent un Higgs leger :
mH = 97+53
GeV
(1) et mH < 194 GeV a 95% CL [3].
36
6
theory uncertainty
∆α(5)
had =
0.02761±0.00036
∆χ2
4
2
0
Excluded
Preliminary
10
2
MH [GeV]
Figure 1.2: 2 = 2 2min a partir d'un t global de l'ensemble des donnees electrofaible
de precision [4], en fonction de mH . La bande bleue (ou grisee, qui suit la courbe des
minimas) indique l'eet des incertitudes theoriques. La zone jaune (ou grisee, avec mH <
114:4 GeV ) est exclue par les recherches directes a LEP2 [5].
La situation actuelle, issue des derniers resultats du LEP2 (caracterise par ps =
209 GeV ) est assez ambigue. En eet, un Higgs avec mH < 114:4 GeV est exclu a 95%
CL mais il y a une legere indication de signal a 1.7 pour le domaine 115 < mH <
118 GeV . Cette estimation vient essentiellement de la collaboration travaillant sur le
detecteur ALEPH qui, a elle seule, mesure un eet a 3[6]. Comme cet eet correspond a
la limite cinematique du LEP2 et qu'une seule collaboration sur les quatres du LEP y est
sensible, il a ete decide de stopper l'experience n 2000 pour pouvoir lancer la construction
du LHC. Pour plus de details a ce niveau on pourra consulter [6].
Les seules avancees recentes viennent du Tevatron qui contraint de mieux en mieux les
proprietes du quark top et du boson W , ce qui permet un meilleur contr^ole des corrections
14
radiatives et donc une meilleure estimation indirecte de la masse du Higgs. Les resultats
les plus recents [7], schematises sur la gure 1.3 et tenant compte des derniers calculs
theoriques, indiquent toujours un Higgs relativement leger (m^eme si le minimum a ete
augmente et passe maintenant a mH 168 GeV ).
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600
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incluant les corrections a deux boucles en fonction de
Figure 1.3: Predictions pour sin2 eff
la masse du Higgs. La bande est obtenue en tenant compte des incertitudes experimentales
a 1. La droite pointillee verticale est la limite d'exclusion de LEP2.
Par ces quelques paragraphes nous avons voulu montrer que l'existence d'un champ
scalaire massif fondamental et de nature quantique n'a toujours pas ete conrmee et pose,
par ailleurs, de serieux problemes a la theorie quantique des champs. De nombreuses solutions, plus ou moins satisfaisantes ont ete proposees, mais une decouverte experimentale
devient indispensable pour faire avancer notre comprehension. A ce niveau, realisons que
de nombreux problemes en cosmologie sont (partiellement) resolus via l'introduction d'un
champ scalaire \classique". On comprend donc que du point de vue de la physique microscopique ces modeles sont au mieux des descriptions eectives de la realite. Cependant,
l'ere de precision en cosmologie dans laquelle nous entrons a present nous laisse esperer une
meilleure comprehension de la nature de ces champs scalaires (ou des lois sous-jacentes)
via l'etude de l'inniment grand.
A present, replongeons-nous dans le monde microscopique et les imperfections du
modele standard. Nous avons vu que ces dernieres sont nombreuses et que les theories
ou modeles alternatifs proposes le sont aussi. Pour de plus amples details je renvoie le
lecteur interesse vers les introductions donnees dans ma these [8], dans la suite je ne decris
que les points essentiels.
Il y a deux methodes dierentes pour tester les manifestations d'une nouvelle physique:
soit on realise un test direct des modeles en tenant compte de l'ensemble des details
15
speciques a chaque modele, soit on essaye d'etablir une approche independante de tout
modele permettant de tester simultanement plusieurs classes de modeles. Mon travail etant
plus phenomenologique que theorique, je me suis surtout penche sur la seconde approche.
Par exemple, un tres grand nombre de theories et modeles prevoient l'existence de nouvelles interactions (e.g. theories de grande unication, supersymetrique ou non, modeles
derives des theories des cordes, modeles de sous-structure). En general, chaque modele
possede des predictions precises sur les couplages entre les nouveaux bosons de jauge et les
particules elementaires du modele standard, fournissant ainsi certaines proprietes globales
telle que les largeurs de desintegration par exemple. En revanche, d'autres proprietes (e.g.
masses, angles de melange) sont souvent des parametres libres qui doivent ^etre xes par
l'experience. Cependant, les contraintes experimentales que l'on obtient dependent, bien
entendu, des proprietes xees par le modele. Ainsi on derive un ensemble de contraintes
pour chaque modele, mais au vu du nombre impressionnant de modeles possibles, on peut
se poser la question de l'interet de ces contraintes ... Une approche modele independante,
bien decrite dans [9] qui reprend directement les travaux pionniers de Fermi sur la description des interactions faibles dans les annees 30, permet de reduire l'ensemble des parametres
du modele a la seule energie caracteristique () du nouveau phenomene. La description
en terme de boson de jauge intermediaire est remplacee par une interaction eective, dite
\de contact",
valable a des energies inferieures a l'echelle d'energie de la nouvelle physique
(i.e. ps ). Concretement, on ajoute un nouveau terme \eectif" au lagrangien du
modele standard, donne, par exemple, par l'expression suivante :
2
Lqq = 2g 2 (LL qLqL:qL qL + RR qRqR:qR qR + 2LR qLqL:qR qR ) (1.2)
qq
ou g est la constante de couplage de la nouvelle interaction et est normalise a g2 = 4. ij
est tel que jij j 1, son signe caracterise le type, destructif ou constructif, des interferences
entre les termes de contact et les amplitudes standards, le type dependant du processus
etudie. qL;R sont les composantes gauche et droite du quark q.
Une attention particuliere a donc ete portee sur les manifestations ou les contraintes que
l'on pourrait obtenir sur ces interactions de contact, que ce soit dans le secteur des quarks
uniquement ou dans le secteur commun aux quarks et aux leptons. Cependant, lorsque
les energies caracteristiques de l'experience et de la nouvelle interaction se rapprochent,
des eets subtils peuvent avoir lieu (e.g. resonance) et une description plus complete des
modeles est preferable. Ainsi nous avons ete amenes a etudier plus en details les manifestations des bosons dits \leptophobes" (bosons de jauge ne couplant qu'aux quarks et
etant ainsi tres dicile a detecter), ou encore de \leptoquarks" (bosons couplant leptons
et quarks directement au sein d'un m^eme vertex). Les premiers ont ete etudies dans le
cadre de l'experience RHIC-Spin avec faisceaux de protons (et neutrons) polarises, qui
est en cours de fonctionnement. Ces resultats sont presentes dans la premiere partie de
ce chapitre. Les seconds ont ete etudies dans le cadre de l'experience HERA, dont un
des projets etait de polariser les faisceaux. Malheureusement, aujourd'hui, le projet a ete
16
abandonne et l'experience arretee, ce qui rend ces etudes obsoletes pour le moment. De
fait, ces etudes seront presentees a la n de ce chapitre.
En parallele a ces travaux, j'ai participe a un travail commun entre les experimentateurs
du CPPM et les theoriciens du CPT sur la phenomenologie de la supersymetrie au LHC. Ce
domaine etant tres vaste, notre approche a ete pragmatique. Nous nous sommes interesses
au processus de production d'un seul quark top (\single top production") qui a l'avantage
de n'^etre sensible qu'a certaines manifestations de nouvelle physique (au premier ordre).
Il est apparu que c'est un processus privilegie pour tester une symetrie propre a la supersymetrie appelee \R parite". En eet, si cette symetrie est brisee des eets nouveaux
devraient ^etre detectes dans la production d'un seul top. Le test de cette symetrie est,
en fait, encore plus important pour la cosmologie car si elle se revele ^etre brisee c'est le
candidat matiere noire issu de la supersymetrie qui disparait .... Cette etude, qui ne fait
absolument pas appel aux eets de polarisation, est presentee dans la deuxieme partie de
ce chapitre.
1.2
Nouvelle Physique aupres de RHIC-Spin
1.2.1 L'experience RHIC-Spin et la crise du spin
Le "Relativistic Heavy Ion Collider" (RHIC) est en cours de fonctionnement depuis 2001
au Brookhaven National Laboratory (BNL-USA), et est voue essentiellement a l'etude
de collisions d'ions lourds de tres hautes energies [10]. Cependant, ce collisionneur fonctionne entre 5 et 10 semaines par an, en mode proton-proton avec faisceaux polarises. La
Collaboration RHIC-Spin a ainsi vu
le jour et essaye actuellement d'augmenter l'energie
caracteristique de l'experience de ps = 250 GeV a ps = 500 GeV . Des eorts tres important sont entrepris pour atteindre les 70% de polarisation par faisceau escompte, ainsi
que la luminosite integree de L = 800 pb 1.
Une autre particularite interessante du RHIC, liee a sa nature de collisionneur d'ions
lourds, est qu'il sera capable d'accelerer des faisceaux de hadrons, autres que des protons,
en particulier des deuterons et des noyaux d'helium 3. L'acceleration de noyaux d' 3He
polarises se revele d'un grand inter^et, car le principe de Pauli nous indique que les spins
des protons sont dans des directions opposees. Ainsi, la direction de la polarisation d'un
noyau d' 3He est, a peu pres, equivalente a celle d'un neutron [11, 12]. Cela nous permet de
supposer que l'on beneciera egalement de faisceaux de neutrons polarises de bonne qualite.
L'inter^et premier du RHIC [13] est la possibilite d'une etude de haute precision de QCD
et de certains phenomenes electrofaibles. En particulier cela va permettre pour la premiere
fois une bonne calibration des distributions partoniques polarisees [14], incluant celle des
gluons, d'une facon radicalement dierente de ce qui a ete realise dans les experiences de
diusion inelastique polarisee. Avant de discuter des possibles eets de nouvelle physique,
je voudrais dire quelques mots sur la crise du spin et les premiers resultats de RHIC-Spin,
17
relativement \inattendus" vu qu'ils conrment que l'on ne comprend pas grand chose a
l'origine du spin du proton et de fait aux aspects (non perturbatifs) de QCD lies au connement.
Le domaine de la physique du spin, controle essentiellement par l'interaction forte a
basse energie, est toujours en pleine eervescence gr^ace aux resultats precis des experiences
de diusion profondement inelastique avec faisceaux de leptons et cibles hadroniques polarises (pour une revue voir [15]) ainsi que ceux recents de RHIC-Spin (e.g. [16]). Le
resultat le plus surprennant est que la contribution des quarks et antiquarks au spin du
proton n'est que d'un quart ! On peut representer ce resultat a l'aide de la regle de somme
du spin du proton [17] :
1 = 1 (Q2) + G(Q2) + L (Q2) + L (Q2) ;
(1.3)
q
g
2 2
et le fait qu'on obtient experimentalement 0:25. Ceci implique qu'une contribution
importante au spin du proton doit venir soit de la polarisation des gluons G(Q2) soit des
moments orbitaux angulaires des quarks Lq (Q2) ou des gluons Lg (Q2). La dependance de
ces moments sur l'echelle d'energie caracteristique de la collison Q a ete incluse explicitement pour rappeler que les distributions partoniques ainsi que leurs moments peuvent
dicilement ^etre predits theoriquement a cause des eets non perturbatifs de QCD. En
revanche, les evolutions avec l'energie de ces diverses quantitees sont parfaitement calculables (e.g. G(Q2) log(Q2) s 1(Q2))[18]. Rappelons que les distributions partoniques
sont denies par :
(1.4)
f + = f++ = f
f = f+ = f +
ou l'invariance des interactions fortes sous la transformation de parite, permet de denir
les distributions ou le parton f a son spin parallele (f +) ou antiparallele (f ) au spin du
hadron parent. On denit alors les distributions de partons :
independantes du spin : f (x; Q2) = f+(x; Q2) + f (x; Q2)
polarisees
: f (x; Q2) = f+(x; Q2) f (x; Q2)
ou f = q; q; g et x represente la fraction d'impulsion emporte par le parton f . Les moments
des distributions partoniques intervenant dans l'eq.(1.3) sont donnes par :
( ) =
Q2
Z 1X
0
q
(q(
x; Q2
) + q(
x; Q2
)) dx
G( ) =
Q2
Z1
0
g(x; Q2) dx: (1.5)
La crise du spin a debutee en 1988 avec les resultats de la collaboration EMC [19] qui
obtint (11 GeV 2) = 0:01 0:29. Depuis, les experiences de diusion profondement
inelastique polarisee de plus en plus precises ont amene la valeur centrale a 0.25 et l'erreur
a quelques pour-cents. On a alors cherche a contraindre G mais la diusion leptonhadron ne donne un acces a g qu'indirectement, a travers l'evolution des distributions ou
via l'etude de processus rares. Jusqu'a l'avenement du RHIC-Spin les contraintes etaient
18
faibles, voire inexistantes. Du point de vue de la theorie, les estimations (sans parler des
calculs) sont extr^emement diciles a realiser car dans le domaine non-perturbatif. A partir
d'un raisonnement base sur la tres forte evolution en Q2 de G, on penchait plut^ot [18, 20]
vers une (tres) forte polarisation des gluons (G > 1:5). En utilisant le \MIT bag model"
ou des \modeles de quarks" non relativistes [21], on obtient des estimations de l'ordre du
spin du proton (G(1 GeV 2) 0:2 0:3 [22, 23]). Tres recemment et pour la premiere
fois une estimation de la dependance en x de g a ete fournie [23].
Vu la profonde incertitude theorique et les resultats experimentaux surprenants, il devenait urgent d'obtenir des mesures de la polarisation des gluons. C'est un des objectifs
principaux du RHIC-Spin [13]. Plusieurs processus, comme la production de photons
("directs"), de jets, de 0 ou encore de quarks lourds, donnent un acces direct a g.
L'observable a utiliser est une double asymetrie de spin qui conserve la symetrie de Parite:
++ +
(1.6)
ALL = ++ +
+
ou est la section ecace du processus considere pour une collision avec deux protons
d'helicites 1 et 2.
Les premiers resultats, presentes sur la gure 1.4, on ete obtenus il y a un peu plus d'un
an, dans le canal de production de 0 pour la collaboration PHENIX, et pour la production
de jets avec STAR. Les predictions de plusieurs modeles phenomenologiques pour g sont
aussi donnes sur la gure 1.4.
On constate qu'une forte polarisation des gluons est exclue. Les resultats sont en accord
avec une polarisation des gluons nulle ou faible. Les gluons ne sont donc pas a l'origine
du spin du proton ! Par consequent, la crise du spin se renforce car il est dicile de
croire que les moments angulaires orbitaux jouent un r^ole important. (Pour une discussion interessante qui a inspiree les lignes precedentes voir [16] ainsi que les deux premiers
chapitres de ma these [8])
A priori, la nouvelle physique n'est pour rien dans cette crise du spin qui devrait trouver
sa source dans notre tres grande diculte a realiser des calculs et a avoir de l'intuition
dans le domaine non perturbatif (de QCD). Cependant, les experiences avec faisceaux
polarises peuvent fournir des contraintes tres interessantes et des informations uniques sur
la structure chirale d'une nouvelle interaction, a la condition de disposer d'une energie et
d'une luminosite susantes. C'est le cas de RHIC-Spin contrairement aux experiences sur
cibles xes.
Par ailleurs, la mise en evidence de nouvelle physique demande la distinction entre un
phenomene non-standard et une uctuation de la prediction standard, liee a la faible connaissance de ces distributions partoniques. La calibration des distributions polarisees au
RHIC [27, 14] nous laisse supposer que ces incertitudes seront fortement reduites dans un
avenir proche. En eet, la collaboration RHIC-Spin apl'intention d'utiliser les runs de 2008,
qui se feront pour la premiere fois a une energie de s = 500 GeV , pour etudier les productions de bosons W qui sont particulierement sensibles aux distributions des quarks et
1
2
19
0
π
ALL
PHENIX Preliminary
0.06
GRSV-max
Run5 Photon Trigger
Run6 Photon Trigger, high pT
0.04
∆g = -g
0.02
GRSV-std
0
-0.02
-0.04
0
∆g = 0
Scaling error of 40%
is not included.
1
2
3
4
5
6
7
8
9
10
pT (GeV/c)
Figure
1.4: Donnees sur l'asymetrie a 2 spins ALL pour la production inclusive de 0 a
ps = 200
GeV mesurees par la collaboration PHENIX [24] (gauche), et pour la production
de jets mesuree par la collaboration STAR [25] (droite), comparees aux predictions NLO
pour plusieurs distributions polarisees des gluons decrites dans [26].
antiquarks. Une fois que ces distributions partoniques seront fortement contraintes il sera
alors possible de se tourner vers le programme de recherche de signaux de nouvelle physique.
Les travaux qui vont suivre (ainsi que ceux presentes dans ma these sur ce sujet la)
ont suscite un vif inter^et au sein de la collaboration RHIC-Spin, ce qui m'a permis de
participer activement aux rencontres de la collaboration et a motiver plusieurs personnes d'etudier ces eets exotiques. Je donnerai en n de section une rapide discussion des
travaux realises dans la communaute depuis la parution des 3 articles qui sont presentes ici.
Avant de detailler
chaque article il faut essayer de comprendre comment l'experience
p
RHIC-Spin avec s = 500 GeV peut ^etre competitive pour decouvrir de nouveaux phenomenes
avec des acceleprateurs possedant des energies plus elevees (SppS au CERN dans les ann
ees
p
1980-90 avec s = 630 GeV , Tevatron au Fermilab actuellement en service avec s =
2 T eV ). Il appara^t que si des leptons sont produits (directement) et etudies dans l'etat
nal (e.g. processus de Drell-Yan), alors le Tevatron fournit les meilleures contraintes car
les processus leptoniques sont "propres" experimentalement. Dans ce cas c'est l'energie qui
prime (avec la luminosite en second plan) dans une optique de decouverte. En revanche,
si le processus est purement hadronique (e.g. production de jets) et donc controle essentiellement par QCD, on recherche des evenements a haute impulsion transverse mais on
est alors confronte, experimentalement, a des problemes de reconstruction ce qui donne de
forts bruits de fonds, et theoriquement, a des incertitudes dans l'estimation des sections
20
ecaces (fortes corrections des ordres superieurs). Ces deux aspects deteriorent fortement
les contraintes susceptibles d'^etre fournies.
C'est pour ce type de processus purement hadroniques que RHIC-Spin est particulierement interessant et plus performant que ses concurrents non polarises plus energetiques. Cette caracteristique vient de la presence de la polarisation qui permet de denir
les nouveaux observables que sont les asymetries de spin, et en particulier l'asymetrie a 1
spin qui viole la Parite1 :
+ (1.7)
AL = +
+
La dierence des sections ecaces au numerateur permet d'eliminer tous les bruits de fonds
associes aux processus conservant la Parite gouvernes par QCD et QED. Le rapport de
sections ecaces devrait permettre de limiter l'impact des corrections d'ordre superieur.
Cependant nous verrons dans notre cas d'etude que cet argument n'est pas correct (voir
la discussion en n de section sur l'estimation NLO des asymetries de spin).
Par consequent, des eets de nouvelle physique pourront ^etre detectes au RHIC-Spin si
et seulement si les nouveaux phenomenes sont purement hadroniques et violent la Parite.
Heureusement, de nombreux modeles theoriques satisfont ces deux criteres. Pour plus de
details sur l'inter^et de l'utilisation de faisceaux polarises pour decouvrir et etudier de la
physique au-dela du modele standard pour divers types de collisionneurs, je conseille le
compte-rendu [29] ou une telle revue m'avait ete demandee.
Le premier article presente ici traite de la manifestation de bosons leptophobes. Des
modeles speciques sont etudies, suivis d'une approche purement phenomenologique. Une
attention particuliere a ete portee a la complementarite entre collisionneurs. Il appara^t
que dans les fen^etres de masse MZ 300 GeV et MZ < 100 GeV seul RHIC-Spin peut
decouvrir de telles particules.
Le deuxieme article a ete redige suite a une demande des responsables de la collaboration
RHIC-Spin qui voulaient connaitre l'impact d'une possible augmentation en energie et/ou
en luminosite de la machine.
En eet, en utilisant au maximum les capacites du RHIC, il est
p
possible d'atteindre s = 650 GeV et L = 20 fb 1. Avec ces nouveaux parametres nous
avons calcule l'amelioration des contraintes en cas de presence de nouvelles interactions de
contact ou de bosons leptophobes. Les resultats obtenus montrent qu'une telle amelioration
de la machine augmente fortement ses capacites de decouverte.
Le dernier article traite des informations que l'on peut obtenir sur les bosons leptophobes en utilisant a la fois des collisions polarisees proton-proton et neutron-neutron (via
la polarisation de noyaux d'He3). La structure chirale de l'interaction, mais aussi et de
facon tres interessante, des informations sur la structure scalaire (type de couplage avec
les bosons de Higgs) peuvent ^etre obtenues.
0
0
1 Dans les articles c'est l'asymetrie a 2 spins violant la Parite AP V
LL
=
++ ++ +
qui est discut
ee, mais il
appara^
t qu'elle est equivalente a l'asym
etrie a 1 spin avec l'inconvenient d'avoir une expression relativement
compliqu
ee lorsque l'on tiend compte des degr
es de polarisation partiels de chaque faisceau (voir [28] pour
les formules et d
etails).
21
1.2.2 Recherche de Z 0 leptophobes
Article publie sous la reference : Phys. Lett. B441 (1998) 376 .
22
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Article publie sous la reference : Phys. Lett. B522 (2001) 89 .
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66
1.2.5 Discussion et prospective
Les analyses proposees precedemment seront realisees prochainement au RHIC. Neanmoins,
ces etudes ont ete realisees a l'ordre le plus bas de la theorie des perturbations. Dans un
contexte hadronique il est clair que les corrections QCD sont tres importantes et doivent
^etre prises en compte. L'utilisation d'asymetries de spin reduit ce probleme comparativement a toute analyse basee sur l'etude de sections ecaces non polarisees. Cependant,
il faut realiser que les contributions non nulles a ces asymetries de spin violant la parite
proviennent d'interferences entre les diagrammes avec echange de gluons et ceux faisant
intervenir l'echange des autres bosons de jauge, et en particulier les neutres (Z 0 et Z 0).
Remarquons que les bosons Z 0 et W fournissent une contribution directe (resonance) ce
qui se traduit par le petit pic a ET 40 45 GeV dans la gure 2 du premier article
presente precedemment. Il en sera de m^eme du Z 0 leptophobe si MZ < 300 GeV . En
d'autres termes, si on neglige les resonances, les contributions dominantes (LO) viennent
a l'ordre O(sW )(des processus a 4 quarks).
A l'ordre suivant O(s2W ) (NLO) on constate que de tres nombreux diagrammes vont
intervenir [30, 31] (voir aussi [32] pour la fastidieuse liste de diagrammes !). En eet, il
faut tenir compte, d'une part, des corrections QCD (O(s)) aux interferences precedentes
(O(sW )), mais aussi d'autre part, des corrections faibles (O(W )) a tous les processus
QCD (O(s2)). Ce dernier type de corrections va faire entrer dans le jeu les gluons. D'autres
corrections a prendre en compte sont celles concernant le denominateur de l'asymetrie.
Le terme dominant est d'ordre O(s2) mais les corrections d'ordre O(s3) peuvent atteindre 60 a 80% d'augmentation (facteur K ) selon l'energie consideree. Ces corrections au
denominateur vont donc avoir un eet important sur la normalisation de l'asymetrie mais
heureusement aucun sur sa forme (car QCD conserve la parite).
Par consequent, au vu de l'explosion du nombre de diagrammes (plus d'un facteur 10),
de la nouvelle contribution des gluons, des eets de denominateur et en realisant que les
corrections NLO font intervenir seulement un facteur s 0:1 supplementaire, il devenait
extr^emement dicle d'estimer intuitivement la veritable importance de ces corrections.
La proximite des mesures et la volonte d'obtenir des resultats aussi precis que possible
imposait donc la necessite de realiser les calculs a l'ordre suivant.
J'ai ainsi discute des dierentes corrections a calculer avec Werner Vogelsang (BNL),
Daniel de Florian (Universite de Buenos Aires) et Marco Stratmann (RIKEN). Finalement
c'est Stefano Moretti et Douglas Ross de l'Universite de Southampton qui ont entrepris
ces calculs [33, 31]. Les principaux resultats pour la production de jets au RHIC [31] sont
donnes sur la gure 1.5.
On constate ainsi que les corrections sur l'asymetrie varient de +100% a -70% selon
l'energie transverse consideree pour le jet ! Cependant, la gure 1.5 donne ces predictions
pour deux ensembles de distributions partoniques polarisees dierents (Ghermann-StirlingA [34] et GRSV-standard [26]) et on peut realiser que les corrections changent beaucoup
d'une parametrisation a l'autre.
0
67
Ceci nous montre donc qu'an d'etudier d'eventuels signaux de nouvelle physique au
RHIC-Spin, il est fondamental de tenir compte des corrections d'ordre superieur mais aussi
de reduire considerablement les incertitudes sur les distributions partoniques, ce qui est
heureusement au programme de la collaboration RHIC-Spin.
Figure 1.5: Corrections a l'ordre O(s2pW ) en fonction de l'energie transverse pour la
production depjets au RHIC-Spin avec s = 600 GeV (courbes s'etendant jusqu'a ET =
300 GeV ) et s = 300 GeV (courbes s'etendant jusqu'a ET = 150 GeV ). Les 4 plots
correspondent a quatre observables distincts : section ecace, asymetrie conservant la
parite, asymetries violant la parite a 1 et 2 spins (label sur les gures), et ceci pour deux
types de distributions partoniques polarisees. Source [31]
68
Recherche de la supersymetrie violant la R parite
au LHC
L'etude qui suit concerne la recherche de signaux supersymetriques violant la R Parite
aupres du collisionneurs LHC dans le canal de production d'un seul top (pp ! tb + X ).
Le LHC est un collisionneur proton-proton qui devrait entrer en fonction tres prochainement (2008). L'epnergie des faisceaux (non-polarises) sera de 7 T eV donnant ainsi un energie
caracteristique ( s = 14 T eV ) superieure d'un ordre de grandeur a celle du collisionneur
actuel le plus puissant. Avec une luminosite integree de L = 10 fb 1 attendue au bout
d'un an dans le cas le plus defavorable, le LHC represente le meilleur espoir de nouvelles
decouvertes pour la communaute.
Le processus de production d'un seul quark top semble prometteur pour tester des
modeles violant les nombres leptonique (L) et baryonique (B), ce qui est le cas des modeles
supersymetriques avec R parite brisee. Un autre avantage de ce processus est que les
predictions du modele standard sont relativement bien etudiees [35, 36]. D'autre part
la contamination du signal par d'autres processus est tres bien simulee par les dierents
generateurs d'evenements.
Article publie sous la reference : Phys. Rev. D61 (2000) 115008.
1.3
69
Single top production at the LHC as a probe of R parity violation.
P. Chiappetta
Centre de Physique Théorique, UPR7061, CNRS-Luminy, Case 907,
F-13288 Marseille Cedex 9, France
A. Deandrea
Theoretical Physics Division, CERN, CH-1211 Geneva 23, Switzerland
E. Nagy and S. Negroni
Centre de Physique des Particules de Marseille, Université de la Méditerranée,
Case 907, F-13288 Marseille Cedex 9, France
G. Polesello
INFN, Sezione di Pavia, via Bassi 6, I-27100 Pavia Italy
J.M. Virey
Institut für Physik, Universität Dortmund, D-44221 Dortmund, Germany and,
Centre de Physique Théorique, UPR7061, CNRS-Luminy, Case 907,
F-13288 Marseille Cedex 9, France and Université de Provence, Marseille, France
(October 1999)
We investigate the potential of the LHC to probe the R parity violating couplings involving the
third generation by considering single top production. This study is based on particle level event
generation for both signal and background, interfaced to a simplified simulation of the ATLAS
detector.
I. INTRODUCTION
The conservation of the baryon B and lepton L number is a consequence of the gauge invariance and renormalizability of the Standard Model. In supersymmetric extensions of the Standard Model, gauge invariance and
renormalizability do not imply baryon and lepton number conservation. We shall consider in what follows the Minimal Supersymmetric Standard Model (MSSM) together with baryon or lepton number violating couplings. These
Yukawa-type interactions are often referred to as R-parity violating couplings. They can mediate proton decay to an
unacceptable level and for this reason a discrete symmetry R was postulated [1] that acts as 1 on all known particles
and as −1 on all the superpartners:
R = (−1)3B+L+2S
(1)
where S is the spin of the particle. In the MSSM with a conserved R-parity the lightest supersymmetric particle
(LSP) cannot disintegrate into ordinary particles and is therefore stable. The superpartners can be produced only in
pairs so that one needs usually to wait for high energy colliders.
In models [2] not constrained by the ad-hoc imposition of R-parity one can still avoid proton decay and the
experimental signatures can be quite interesting: single production of supersymmetric particles and modification of
standard decays and cross-sections due to the exchange of these sparticles, which could be observed at lower energies
compared to the R-parity conserving model. In the following we shall investigate top quark production taking into
account R-parity violating effects. The top quark being heavy with a mass close to the electroweak symmetry breaking
scale, it is believed to be more sensitive to new physics than other quarks. The mechanism we plan to study is single
top quark production at LHC, which is complementary to top quark pair production and reliably well known in the
Standard Model.
Two basic ways to probe new physics can be investigated. The first one is a model independent analysis, in which
the effects of new physics appear as new terms in an effective Lagrangian describing the interactions of the third
family with gauge bosons and Higgs [3–5]. The effects due to the interactions between quarks and gauge bosons will
be visible at LEP2, e+ e− next linear colliders and the Tevatron whereas dimension 6 CP violating operators affect
the transverse polarisation asymmetry of the top quark. The second way is to consider a new theory which contains
1
the Standard Model at low energies. A possible framework is supersymmetry. In the Minimal Supersymmetric Model
with R parity conservation, the single top production at Tevatron is enhanced by a few percent due to gluino, squarks,
higgs, charginos and neutralinos corrections, the magnitude being sensitive to tan β [6]. The decays t → cV with
V = g, Z, γ, which are small in magnitude in the Standard Model (BR ' 10−10 − 10−12 ), may be enhanced by a few
orders of magnitude in the MSSM [7]. If the stop and the charged Higgs are light enough new top decays are possible
[8]. Our purpose is to investigate the effects of R parity violation. The superpotential contains three types of new
terms:
00
W6R = λijk Li Lj Ēk + λ0ijk Li Qj D̄k + λijk Ūi D̄j D̄k
(2)
the first two terms violating the leptonic number and the last the baryonic one. Here L and E are isodoublet and
isosinglet lepton, Q and D are isodoublet and isosinglet quark super-fields, the indices i, j and k run for the three
lepton and quark families. In the following we shall assume that R-parity violation arises from one of these terms
only.
The feasibility of single top quark production via squark and slepton exchanges to probe several combinations of R
parity violating couplings at hadron colliders has been studied [9–11]. The LHC is better at probing the B violating
couplings λ00 whereas the Tevatron and the LHC have a similar sensitivity to λ0 couplings. We perform a complete
and detailed study including for the signal all channels using a Monte Carlo generator based on Pythia 6.1 [12], taking
into account all the backgrounds and including the ATLAS detector response using ATLFAST 2.0 [13].
The paper is organised as follows. Section II is devoted to an evaluation of the different subprocesses contributing
to single top production (standard model, squark, slepton and charged Higgs exchanges). The potential of the LHC
to discover or put limits on R-parity violating interactions is given in section III.
II. SUBPROCESSES CONTRIBUTING TO SINGLE TOP PRODUCTION
The R-parity violating parts of the Lagrangian that contribute to single top production are:
00
L6R = λ0ijk ẽiL d¯kR ujL − λijk (d˜kR ūiL djL + d˜jR (d¯kL )c uiL ) + h.c.
(3)
The superscript c corresponds to charge conjugation. There are altogether 27 and 9 λ 0ijk and λ00ijk Yukawa couplings,
00
00
respectively. The most suppressed couplings are λ0111 ≤ 3.5 × 10−4 , λ0133 ≤ 7 × 10−4 , λ112 ≤ 10−6 , λ113 ≤ 10−5 (for
bounds see [14]). In order to fix the kinematical variables, the reaction we consider is
ui (p1 ) + dj (p2 ) → t(p3 ) + b(p4 ) ,
(4)
the pk being the 4-momenta of the particles and the indices i and j refer to the generations of the u and d-type quarks.
We first discuss valence-valence (VV) or sea-sea (SS) subprocesses (this notation refers to the proton-proton collisions
at the LHC, but the calculation is valid in general). The SM squared amplitude due to W exchange in û-channel 1 is
suppressed by the Kobayashi-Maskawa matrix elements Vui b Vtdj :
VV 2
4
2
2
|MW
W | = g |Vui b | |Vtdj |
1
p1 · p2 p3 · p4 ,
(û − m2W )2 + m2W Γ2W
(5)
where g, m and Γ denote the weak coupling constant, the mass and the width of the exchanged particle. The H ±
exchange in û-channel is included in the calculation but numerically suppressed by the quark masses and the mixing
matrix elements for the charged Higgs sector Kui b Ktdj (under the assumption K = V ):
VV
2
|MH
=
±H± |
1
g4
|Kui b |2 |Ktdj |2
16 m4W
(û − m2H ± )2 + m2H ± Γ2H ±
2
2
[(vbu
+ a2bui ) p1 · p4 + (vbu
− a2bui ) mb mui ][(vd2j t + a2dj t ) p2 · p3 + (vd2j t − a2dj t ) mdj mt ],
i
i
vud and aud are respectively the vector and axial vector couplings of H ± to quarks:
1
The ”hat” symbol refers to the usual Mandelstam variables for the process at the parton level
2
(6)
vud = md tan β + mu cot β
aud = md tan β − mu cot β
(7)
The interference term between the W and H ± is:
VV
2 Re(MW
H± ) = −
(û − m2W )(û − m2H ± ) + mW ΓW mH ± ΓH ±
g4
|Vui b | |Vtdj | |Kui b | |Ktdj |
2
8 mW
[(û − m2W )2 + m2W Γ2W ][(û − m2H ± )2 + m2H ± Γ2H ± ]
× [(vbui + abui )(vdj t + adj t ) mb mdj p1 · p3 + (vbui + abui )(vdj t − adj t ) mb mt p1 · p2
+ (vbui − abui )(vdj t + adj t ) mui mdj p3 · p4 + (vbui − abui )(vdj t − adj t ) mui mt p2 · p4 ].
(8)
The scalar slepton exchange in û-channel is taken into account but appears to be suppressed within our assumptions
on the λ0 couplings (see below). It reads:
02
|MẽVkVẽk |2 = λ02
ki3 λk3j
L L
1
p1 · p4 p2 · p3 .
(û − mẽ2k )2 + mẽ2k Γẽ2k
L
L
(9)
L
The interference term between scalar slepton and W reads:
VV
2 Re(MW
)
ẽk
L
= −g
2
|Vui b | |Vtdj | λ0ki3
λ0k3j
(û − m2W )(û − mẽ2k ) + mW ΓW mẽk Γẽk
L
L
L
[(û − m2W )2 + m2W Γ2W ][(û − mẽ2k )2 + mẽ2k Γẽ2k ]
L
L
md mb p 1 · p 3 .
(10)
L
The interference term between scalar slepton and H ± , which is suppressed, reads:
2 Re(MẽVkVH ± )
L
(û − m2H ± )(û − mẽ2k ) + mH ± ΓH ± mẽk Γẽk
g2
L
L
0
0
L
|Kui b | |Ktdj |
=
λ λ
4 m2W ki3 k3j
[(û − m2H ± )2 + m2H ± Γ2H ± ][(û − mẽ2k )2 + mẽ2k Γẽ2k ]
L
L
L
× [(vbui + abui )(vdj t + adj t ) p1 · p4 p2 · p3 + (vbui + abui )(vdj t − adj t ) mdj mt p1 · p4
+ (vbui − abui )(vdj t + adj t ) mui mb p2 · p3 + (vbui − abui )(vdj t − adj t ) mui mdj mb mt ].
(11)
The down type squark exchange in ŝ-channel squared amplitude is dominant and given by:
|Md̃VkVd̃k |2 =
R R
00
00
1
4
2
2
16 λijk
λ33k
p1 · p2 p3 · p4 .
3
(ŝ − m2d̃k )2 + m2d̃k Γ2d̃k
R
R
(12)
R
The corresponding interference terms are:
VV
2 Re(MW
)
d̃k
R
(û − m2W )(ŝ − m2d˜k ) + mW ΓW md̃k Γd̃k
00
00
2
R
R
2
R
p1 · p2 p3 · p4 ,
= − 8 g |Vui b | |Vtdj | λijk λ33k
3
[(û − m2W )2 + m2W Γ2W ][(ŝ − m2d̃k )2 + m2d̃k Γ2d˜k ]
R
R
(13)
R
and:
2 Re(Md̃VkVH ± )
R
(û − m2H ± )(ŝ − m2d̃k ) + mH ± ΓH ± md̃k Γd̃k
00
00
2 g2
R
R
R
=
λ λ
|Kui b | |Ktdj |
3 2 m2W ijk 33k
[(û − m2H ± )2 + m2H ± Γ2H ± ][(ŝ − m2d̃k )2 + m2d˜k Γ2d̃k ]
R
R
R
× [(vbui + abui )(vdj t + adj t ) mb mdj p1 · p3 + (vbui + abui )(vdj t − adj t ) mb mt p1 · p2
+ (vbui − abui )(vdj t + adj t ) mui mdj p3 · p4 + (vbui − abui )(vdj t − adj t ) mui mt p2 · p4 ].
(14)
Let us now take into account the subprocesses involving valence-sea (VS) quarks. The SM squared amplitude due
2
to W exchange in the ŝ-channel, being proportional to (Vui dj Vtb ) is dominant for quarks of the same generation. It
reads:
VS 2
4
2
2
|MW
W | = g |Vui dj | |Vtb |
1
p1 · p4 p2 · p3 .
(ŝ − m2W )2 + m2W Γ2W
(15)
The charged Higgs contribution in the ŝ-channel is suppressed by the quark masses of the initial state. The squared
amplitude is:
2
VS
=
|MH
±H± |
1
g4
|Kui dj |2 |Ktb |2
16 m4W
(ŝ − m2H ± )2 + m2H ± Γ2H ±
2
2
+ a2bt ) p3 · p4 − (vbt
− a2bt ) mb mt ].
× [(vd2j ui + a2dj ui ) p1 · p2 − (vd2j ui − a2dj ui ) mdj mui ][(vbt
3
(16)
The interference term between W and H ± is:
VS
2 Re(MW
H± ) = −
(ŝ − m2W )(ŝ − m2H ± ) + mW ΓW mH ± ΓH ±
g4
|V
|
|V
|
|K
|
|K
|
u
d
tb
u
d
tb
i
j
i
j
8 m2W
[(ŝ − m2W )2 + m2W Γ2W ][(ŝ − m2H ± )2 + m2H ± Γ2H ± ]
× [(vdj ui + adj ui )(vbt + abt ) mb mdj p1 · p3 − (vdj ui + adj ui )(vbt − abt ) mdj mt p1 · p4
− (vdj ui − adj ui )(vbt + abt ) mb mui p2 · p3 + (vdj ui − adj ui )(vbt − abt ) mui mt p2 · p4 ].
(17)
Concerning R parity violating terms, slepton exchange in ŝ-channel and down type squark exchange in the û-channel
contribute:
02
|MẽVkSẽk |2 = λ02
kij .λk33
L
L
1
p1 · p2 p3 · p4
(ŝ − mẽ2k )2 + mẽ2k Γẽ2k
L
|Md̃VkSd̃k |2
R R
L
L
00
00
1
4
2
2
λ3jk
p1 · p4 p2 · p3 .
= 16 λi3k
3
(û − m2d̃k )2 + m2d̃k Γ2d̃k
R
R
(18)
R
The interference terms involving the scalar lepton are:
0
0
VS
2
2 Re(MW
ẽk ) = − g |Vui dj | |Vtb | λkij λk33
L
(ŝ − m2W )(ŝ − mẽ2k ) + mW ΓW mẽk Γẽk
L
L
L
[(ŝ − m2W )2 + m2W Γ2W ][(ŝ − mẽ2k )2 + mẽ2k Γẽ2k ]
L
L
md mb p 1 · p 3 ,
(19)
L
and:
2 Re(MẽVkSH ± )
L
(ŝ − m2H ± )(ŝ − mẽ2k ) + mH ± ΓH ± mẽk Γẽk
g2
L
L
0
0
L
=
λ λ
|Kui dj | |Ktb |
4 m2W kij k33
[(ŝ − m2H ± )2 + m2H ± Γ2H ± ][(ŝ − mẽ2k )2 + mẽ2k Γẽ2k ]
L
L
L
× [(vdj ui + adj ui )(vbt + abt ) p1 · p2 p3 · p4 − (vdj ui + adj ui )(vbt − abt ) mt mb p1 .p2
− (vdj ui − adj ui )(vbt + abt ) mdj mui p3 · p4 + (vdj ui − adj ui )(vbt − abt ) mui mdj mb mt ].
(20)
The interference terms involving the scalar quark are:
VS
) = −
2 Re(MW.
d̃k
R
(ŝ − m2W )(û − m2d̃k ) + mW ΓW md̃k Γd̃k
00
00
2
R
R
R
8 g 2 |Vui dj | |Vtb | λi3k λ3jk
p1 · p4 p2 · p3
3
[(ŝ − m2W )2 + m2W Γ2W ][(û − m2d˜k )2 + m2d̃k Γ2d˜k ]
R
R
(21)
R
and:
2 Re(Md̃VkS.H ± ) =
R
(ŝ − m2H ± )(û − m2d̃k ) + mH ± ΓH ± md̃k Γd̃k
00
00
2 g2
R
R
R
λ
λ
|K
|
|K
|
ui d j
tb
3 2 m2W i3k 3jk
[(ŝ − m2H ± )2 + m2H ± Γ2H ± ][(û − m2d̃k )2 + m2d˜k Γ2d̃k ]
R
R
R
× [(vdj ui + adj ui )(vbt + abt ) mb mdj p1 · p3 − (vdj ui + adj ui )(vbt − abt ) mdj mt p1 · p4
− (vdj ui − adj ui )(vbt + abt ) mb mui p2 .p3 + (vdj ui − adj ui )(vbt − abt ) mui mt p2 .p4 ].
(22)
The dominant terms are the squared amplitude due to ẽ exchange, and for initial quarks of the same generation
˜ The result is sensitive to the interference term only if the product of λ00
(i = j), the interference between W and d.
couplings is large (around 10−1 ). For subprocesses involving quarks of different generations in the initial state the
situation is more complex and all amplitudes have to be taken into account.
The resonant ŝ-channel processes have been studied in [11], for first family up and down quarks. For the B-violating
couplings, the study of ŝ-channels cd → s̃ and cs → d˜ can also be found in [11]. The û-diagram has been studied at
the Tevatron for the first family of up and down quarks [10].
In the present note we have improved previous calculations for LHC because we have included all contributions
to single top production. Since the dominant terms are those considered in the literature, our complete evaluation
validates the approximations done in previous papers.
III. DETECTION OF SINGLE TOP PRODUCTION THROUGH R-PARITY VIOLATION AT THE LHC
We have carried out the feasibility study to detect single top production through R-parity violation at the LHC by
measuring the lνbb final state using the following procedure.
4
First, we have implemented the partonic 2 → 2 cross sections calculated using Eqs. (5)–(22) in the PYTHIA
event generator. Providing PYTHIA with the flavour and the momenta of the initial partons using a given parton
distribution function (p.d.f.)2 it then generates complete final states including initial and final state radiations and
hadronization.
The generated events were implemented in ATLFAST to simulate the response of the ATLAS detector. In particular,
isolated electrons, photons were smeared with the detector resolution in the pseudo-rapidity range of |η| < 2.5. In
the same way and the same η region the measured parameters of the isolated and non-isolated muons were simulated.
Finally, a simple fixed cone algorithm (of radius R = 0.4) was used to reconstruct the parton jets. The minimum
transverse energy of a jet was set at 15 GeV. According to the expected b-tagging performance of the ATLAS detector
[15] for low luminosity at the LHC we have assumed a 60% b-tag efficiency for a factor 100 of rejection against light
jets.
The same procedure was applied to the SM background with the exception that we used besides PYTHIA also the
ONETOP [16] event generator.
The integrated luminosity for one year at low luminosity at the LHC is taken to be 10 fb −1 .
The number of signal events depends on the mass and the width of the exchanged sparticle, and on the value of the
Yukawa couplings (see Section II). We assume that only one type of Yukawa coupling is nonzero, i.e. either sleptons
(λ0 6= 0) or squarks (λ00 6= 0) are exchanged. The width of the the exchanged sparticle is a sum of the widths due to
R-parity conserving and R-parity violating decays:
Γtot = ΓR + ΓR/
(23)
where ΓR/ is given by
2
2 2
(λ00ijk )2 (Mq̃Ri − Mtop )
2π
Mq̃3i
(24)
2
2 2
3(λ0ijk )2 (Ml̃iL − Mtop )
−→ q q̄ ) =
16π
Ml̃3i
(25)
i
ΓR/(q̃R
−→ q j q k ) =
R
for the squarks, and it is given by
i
ΓR/(l̃L
j k
L
for the sleptons. The number of signal events depends also on the flavour of the initial partons through their p.d.f. In
Table I we display the total cross section values for different initial parton flavours in the case of exchanged squarks of
mass of 600 GeV and of R-parity conserving width ΓR = 0.5 GeV. We took for all λ00 = 10−1 , which yields a natural
width of the squark which is smaller than the experimental resolution. Table II contains the same information for
slepton exchange ( λ0 = 10−1 , for a slepton of mass of 250 GeV and a width of ΓR = 0.5 GeV). Other processes are
not quoted because the small value of the limits of their couplings prevents their detection.
In order to study the dependence of the signal on the mass and the width of the exchanged particle we have fixed
the couplings to 10−1 and have chosen three different masses for the exchanged squarks: 300, 600 and 900 GeV,
respectively. For each mass value we have chosen two different ΓR : 0.5 and 20 GeV, respectively. For the first case
ΓR/ dominates, whereas in the last one, when Γtot ≈ ΓR , the single top-production cross section decreases by a factor
∼ 10. We have considered here the ub parton initial state, since this has the highest cross section value. Besides,
we have also generated events with a cd initial partonic state and an exchanged s̃-quark of mass of 300 GeV, for
comparison with the simulation presented in Ref. [11].
In order to study the dependence on the parton initial state we have fixed the mass of the exchanged squark to 600
GeV and its width with ΓR = 0.5 GeV and varied the initial state according to the first line of Table I.
Finally, for the exchanged sleptons we have studied only one case, namely the ud¯ initial state with a mass and
width of the exchanged slepton of 250 GeV and 0.5 GeV, respectively. In each case we have generated about 10 5
signal events.
The different types of background considered are listed in Table III together with their estimated cross sections.
The irreducible backgrounds are single top production through a virtual W (noted W ∗ ), or through W -gluon fusion.
2
We have used the CTEQ3L p.d.f.
5
W -gluon fusion is the dominant process (for a detailed study see [17]). A W bb final state can be obtained either in
direct production or through W t or tt̄ production. Finally, the reducible background consists of W -nj events where
two of the jets are misidentified as b-jets.
We have used the ONETOP [16] event generator to simulate the W -gluon fusion process. For the other backgrounds
we have used PYTHIA. We have generated from one thousand (W ∗ ) to several million events (tt̄) depending on the
importance of the background.
The separation of the signal from the background is based on the presence of a resonant structure of the tb final
state in the case of the signal. The background does not show such a structure as it is illustrated in Fig 1.
In the process to reconstruct the tb final state first we reconstruct the top quark. The top quark can be reconstructed
from the W and from one of the b-quarks in the final state, requiring that their invariant mass satisfy
150 ≤ MW b ≤ 200GeV.
The W is in turn reconstructed from either of the two decay channels:
W → ud¯
W → lν.
Here we have considered only the latter case which gives a better signature due to the presence of a high p t lepton
and missing energy. The former case suffers from multi-jets event backgrounds. As we have only one neutrino, its
longitudinal momentum can be reconstructed by using the W and top mass constraints. The procedure used is the
following :
- we keep events with two b-jets of pt ≥ 40 GeV, with one lepton of pt ≥ 25 GeV, with Etmiss ≥ 35 GeV and with a
jet multiplicity ≤ 3,
- we reconstruct the longitudinal component (pz ) of the neutrino by requiring Mlν = MW . This leads to an equation
with twofold ambiguity on pz .
- More than 80% of the events have at least one solution for pz . In case of two solutions, we calculate Mlνb for each
of the two b-jets and we keep the pz that minimises |Mtop − Mlνb |.
- we keep only events where 150 ≤ Mlνb ≤ 200 GeV.
Next, the reconstructed top quark is combined with the b quark not taking part in the top reconstruction. An
example of the invariant mass distribution of the tb final state is shown in Fig. 2.
In order to reduce the the tt̄ background to a manageable level, we need to apply a strong jet veto on the third jet
by requiring that its pt should be ≤ 20 GeV.
The invariant mass distribution of the tb final state after having applied this cut is shown in Fig. 3. The signal to
background ratio is clearly increased in comparison to Fig. 2.
Once an indication for a signal is found, we count the number of signal (Ns ) and background (Nb ) events in an
interval corresponding to 2 standard deviations around the signal peak for an integrated luminosity of 30 fb −1 . Then
we rescale the signal peak by a factor α such that
p
Ns / Nb = 5.
By definition the scale-factor α determines the limit of sensitivity for the lowest value of the λ 00 (λ0 ) coupling we can
test with the LHC:
√
λ00ijk · λ00lmn ≤ 0.01 · α.
In Table IV we show the limits obtained for the combinations of λ00132 λ00332 for different masses and widths of the
exchanged s̃-quark. Also shown are the current limit assuming assuming a mass for m̃ f = 100 GeV, the number
of signal and background events, as well as the experimentally observable widths of the peak (Γ exp ). In Fig. 4 we
compare our results with those obtained in Ref. [11] for ms̃ = 300 GeV, and a cd initial state, using parton-level
simulation. We ascribe the lower efficiency of this analysis to the more detailed and realistic detector simulation
employed.
In Table V we compile the sensitivity limit of the bilinear combination of the different Yukawa couplings one can
obtain after 3 years of LHC run with low luminosity, if the exchanged squark has a mass of 600 GeV. For its width
we consider ΓR = 0.5 GeV and a component ΓR/ given by Eq.(24).
For the exchanged sleptons (cf Table II) we have calculated the sensitivity limit of the bilinear combination of
the different Yukawa couplings only for the most favourable case, i.e. for the ud¯ partonic initial state. We obtain
6
4.63×10−3 for the limits on λ011k λ0k33 (in comparison with the limit of 2.8×10−3 obtained by Oakes et al.). For those
cases where the exchanged squark (slepton) might be discovered at the LHC we have made an estimate on the precision
one can determine its mass. For this purpose, we have subtracted the background under the mass peak and fitted a
Gaussian curve on the remaining signal. This procedure is illustrated in Fig. 5 for the case of 600 GeV squark mass
and ub partonic initial state. For the assumed value of the coupling constant, the error on the mass determination is
dominated by the 1% systematic uncertainty on the jet energy scale in ATLAS [15].
IV. CONCLUSIONS
We have studied single top production through R-parity violating Yukawa type couplings, at the LHC.
We have considered all 2 → 2 partonic processes at tree-level, including interference terms. The calculated 2 → 2
partonic cross sections have been implemented in PYTHIA to generate complete particle final states. A fast particle
level simulation was used to obtain the response of the ATLAS detector. We have taken into account all important
SM backgrounds.
We have studied the signal-to-background ratio as a function of the initial partonic states, the exchanged sparticle
mass and width, and of the value of the Yukawa couplings.
At the chosen value of the coupling constants (∼ 10−1 ), significant signal-to-background ratio was obtained only in
the ŝ-channel, in the tb (lνbb) invariant mass distribution, around the mass of the exchanged sparticle, if
(i) the exchanged sparticle is a squark, and
(ii) its width due to R-parity conserving decay
√is of the order of a GeV.
In this case we obtain a significance of S = Ns / Nb > 5 for the whole mass range investigated (300 – 900 GeV) for
an integrated luminosity of 30 fb−1 . This means, that squarks (d˜ or s̃) with narrow width might be discovered at the
LHC. The experimental mass resolution would permit to measure the squark mass in this case with a precision of
∼ 1%.
Conversely, if no single top production above the SM expectation is observed at the LHC, after 3 years of running at
low luminosity, the experimental limit on the quadratic combination of the λ00 couplings can be lowered by at least one
order of magnitude, for narrow width squarks. In the case of the exchanged sleptons significant signal-to-background
ratio can be obtained in the case of ud¯ partonic initial state, i.e. for the combination of the λ011k λ0k33 couplings. Due
to the lower rate, as compared to squark exchange, in the absence of a signal the current limit can be improved only
by a factor 2. The difference between the significance in our study and the one in Ref. [11] can be explained by the
different degree of detail in the simulation process.
ACKNOWLEDGEMENTS.
We thank S. Ambrosanio and S. Lola for useful comments on the manuscript. A.D. acknowledges the support of
a “Marie Curie” TMR research fellowship of the European Commission under contract ERBFMBICT960965 in the
first stage of this work. J.-M. V. thanks the “Alexander von Humboldt Foundation” for financial support.
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Phys. Rev. D56 (1997) 3100; G. Couture, M. Frank and H. Konig, Phys. Rev. D56 (1997) 4213; G.M. de Divitiis, R.
Petronzio and L. Silvestrini, Nucl. Phys. B504 (1997) 45.
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C.P. Yuan, Phys. Lett. B367 (1996) 188; J. Guasch and J. Sola, Phys. Lett. B416 (1998) 353.
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[13] E.Richter-Wa̧s, D.Froidevaux, L.Poggioli, ATLAS Internal Note Phys-No-79, 1996.
[14] G. Bhattacharyya, Nucl. Phys. Proc. Suppl. 52A (1997) 83; H. Dreiner, hep-ph/9707435, in Perspectives on Supersymmetry, ed. G. Kane, World Scientific; Report of the R-parity group of GDR SUSY, hep-ph/9810232, available at
http://www.in2p3.fr/susy/ and references therein.
[15] The ATLAS Collaboration, Detector and Physics Performance Technical Design Report, CERN/LHCC/99-14, ATLAS
TDR 14, 25 may 1999.
[16] Onetop, C.P. Yuan, D. Carlson, S. Mrenna, Barringer, B. Pineiro, R. Brock,
http://www.pa.msu.edu/˜ brock/atlas-1top/EW-top-programs.html
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Events/bin/30 fb
-1
FIGURES
6000
4000
2000
0
0
250
500
750
1000
mbblν (GeV)
Fig. 1 - Invariant mass distribution of lνbb for the backgrounds after three years at LHC at low luminosity. The t t̄
background dominates (dashed histogram).
8
-1
Events/bin/30 fb
4000
3000
2000
1000
0
0
250
500
750
1000
mbblν (GeV)
Events/bin/30 fb
-1
Fig. 2 - Invariant mass distribution of lνbb combination for the signal and backgrounds (dashed histogram) after
three years of LHC run at low luminosity. The signal corresponds to an exchanged s̃-quark of 600 GeV mass and 0.5
GeV width. The initial partons are ub and the λ00 couplings are set to 10−1 .
1000
750
500
250
0
0
250
500
750
1000
mbblν (GeV)
Fig. 3 - Invariant mass distribution of lνbb for the signal and backgrounds (dashed histogram) after three years at
LHC at low luminosity after having applied the cuts.
9
λ”212λ”332
0.06
▲ - Oakes et al.
0.05
■ - Oakes cuts
● - Our analysis
0.04
0.03
0.02
0.01
10
-1
1
Γ⁄msquark (GeV)
Events/bin/30 fb
-1
Fig. 4 - Sensitivity limits for the values of the λ00212 λ00332 Yukawa couplings we obtain for the cd initial state at the
LHC after 1 year with low luminosity, for an exchanged s̃-quark of mass of 300 GeV (circles). The result obtained by
Oakes et al., is also shown (triangles). The squares indicate a result obtained by applying the cuts used by Oakes et
al. on our sample.
200
m = 600.3 GeV
150
100
50
0
400
600
800
1000
1200
mbblν (GeV)
Fig. 5 - The background subtracted mass distribution fitted with a Gaussian in case of an exchanged s̃-quark of 600
GeV for a ub initial parton state. It corresponds to 3 years of LHC run with low luminosity.
10
TABLES
Initial partons
Exchanged particle
Couplings
Cross section in pb
cd
s̃
λ00212 λ00332
3.98
cs
d̃
λ00212 λ00331
1.45
ub
s̃
λ00132 λ00332
5.01
cb
d˜
λ00231 λ00331
s̃
λ00232 λ00332
0.659
TABLE I. Total cross-section in pb for squark exchange in the ŝ-channel for a squark of mass of
600 GeV assuming ΓR = 0.5 GeV.
Initial partons
Couplings
Cross section in pb
ud̄
λ011k λ0k33
7.05
us̄
λ012k λ0k33
4.45
cd̄
λ021k λ0k33
2.31
cs̄
λ022k λ0k33
1.07
ub̄
λ013k λ0k33
2.64
cb̄
λ023k λ0k33
0.525
TABLE II. Total cross-section in pb for slepton exchange in the ŝ-channel for a slepton of mass
of 250 GeV assuming ΓR = 0.5 GeV.
Background
W∗
gluon fusion
Wt
tt̄
W bb
W jj
σ × BR (pb)
2.2
54
17
246
66.6
440
TABLE III. σ× Branching Ratio for backgrounds.
ms̃ (GeV)
ΓR (GeV)
Ns
Nb
Γexp (GeV)
Limits on λ00 × λ00
300
0.5
20
6300
250
4920
5640
24.3
30.5
2.36×10−3 1.21×10−2
600
0.5
20
703
69
558
1056
37.5
55.6
4.10×10−3 1.51 ×10−2
900
0.5
20
161
22
222
215
55.4
62.1
6.09×10−3 2.09×10−2
TABLE IV. Sensitivity limits for the values of the λ00132 λ00332 Yukawa couplings for an integrated
luminosity of 30 fb−1 . For the other quantities see the text. Current limit is 6.25×10−1 .
Initial partons
Exchanged particle
Couplings
Ns
Nb
Γexp (GeV)
Limits on λ00 × λ00
cd
s̃
λ00212 λ00332
660
cs
d˜
λ00212 λ00331
236
38.5
4.26×10−3
31.3
7.08×10−3
ub
s̃
λ00132 λ00332
703
558
37.5
4.1×10−3
cb
d̃
λ00231 λ00331
s̃
λ00232 λ00332
96
40.1
1.11×10−2
TABLE V. Sensitivity limit of the Yukawa couplings for an exchanged squark of mass of 600
GeV assuming ΓR = 0.5 GeV, for an integrated luminosity of 30 fb−1 . Current limit is 6.25×10−1 .
11
1.4 Nouvelle Physique aupres de HERA polarise
Le collisionneur HERA etait dedie a l'etude de la diusion profondement inelastique via
des pcollisions entre protons et electrons (ou positrons). L'energie caracteristique etait
de s = 300 GeV et la luminosite integree a ni par atteindre quelques centaines de
pb 1 ( 0:5 fb 1 ). La polarisation des faisceaux de positrons et d'electrons pouvait ^etre
obtenue sans trop de diculte. En revanche, le faisceau de proton etait non polarise mais
il a ete considere tres serieusement de le polariser lorsque les responsables du collisionneur
et des experiences pensaient au futur d'HERA. J'ai ainsi ete contacte par Albert de Roeck
(DESY/CERN) qui m'a demande de realiser des etudes sur l'inter^et de polariser le faisceau
de protons pour analyser d'eventuels signaux de nouvelle physique. Plus tard, Yves Sirois
(Polytechnique) et Abbhay Deshpande (Yale/BNL) m'ont demande des etudes similaires
pour des projets, aujourd'hui avortes, dep collisionneurs ep avec dieprentes energies, tels
que TESLAxHERA qui aurait pu avoir s = 1 T eV ou eRHIC avec s = 100 GeV mais
une tres forte luminosite.
Les types de nouvelle physique testables a HERA (et autres collisionneurs ep) doivent
faire intervenir de nouveaux couplages entre electrons et quarks. Ces m^eme modeles peuvent induire des eets dans le processus de Drell-Yan dans le cadre des collisions du Tevatron (pp ! l+l + X ). Au vu de la tres grande dierence d'energie entre les deux machines,
il est assez clair que le Tevatron avait la plus grande chance de decouverte. Une etude plus
attentive a montre que les contraintes etaient plut^ot complementaires et a peu pres comparables mais qu'il fallait tenir compte aussi d'autres resultats comme les limites venant du
LEP ou de la violation de parite dans les atomes de Cesium. Par consequent, la question la
plus interessante pour HERA n'etait pas ce qui pouvait y ^etre decouvert mais plut^ot, en cas
de decouverte d'un nouveau couplage electron-quark aupres d'une quelconque experience,
quelles informations auraient pu ^etre obtenues a partir des divers observables mesurables
a HERA.
Les modeles que j'ai consideres sont les interactions de contact entre electrons et quarks
ainsi que les nouvelles interactions dues a la presence de leptoquarks. En ce qui concerne
les interactions de contact, un article a ete publie en 1999 [28] mais les principaux resultats
ont ete donnes dans ma these. Par consequent, je ne reporte pas cette analyse dans cette
habilitation.
L'article qui suit traite des possibles manifestations de leptoquarks. Les leptoquarks
trouvent une origine theorique dans les theories de grande unication ou ils sont associes a des bosons de jauge (leptoquarks vectoriels), dans les theories supersymetriques
violant la R parite ou ils sont associes aux super-partenaires des quarks (leptoquarks
scalaires/squarks), et enn dans certains modeles de sous-structure. L'approche suivie est
purement phenomenologique et ne tiens compte que de l'invariance de jauge. Le resultat
important obtenu est que la polarisation des faisceaux permet une identication de la nature du leptoquark, ce qui semble beaucoup plus dicile voire impossible sans polarisation.
Malheureusement, ces etudes sont obsoletes car le collisionneur HERA ne fonctionne
81
plus. Il faut egalement souligne qu'aucun signal n'a indique serieusement la presence d'une
nouvelle interaction entre electrons et quarks jusqu'a present.
Article publie sous la reference : Eur. Phys. J. C14 (2000) 165.
82
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—ƒè
Chapitre 2 : Determination des
parametres cosmologiques et
proprietes de l'energie noire
Fin 2001, j'ai entrepris avec Pierre Taxil une reconversion thematique. La premiere annee
je me suis interesse a la physique des astroparticules, et en particulier a la physique des neutrinos ou plut^ot des telescopes a neutrinos (ANTARES, AMANDA, ICECUBE), ainsi qu'a
la problematique des rayons cosmiques d'extr^emement haute energie. En parallele, l'ancien
groupe ALEPH (LEP) du CPPM etudiait de plus en plus la physique et la cosmologie associes aux supernovae de type Ia (SNIa) considerees comme chandelles standard. Il est
apparu que les perspectives en cosmologie etaient tres allechantes, et c'est ainsi qu'a partir
de 2002/2003 je me suis concentre sur la cosmologie en liaison avec des experimentateurs.
Le moteur de mes recherches est l'etude des proprietes de l'energie noire, la composante
principale de l'Univers qui est a la source de l'acceleration recente de l'expansion et qui
n'est plus vraiment remise en cause depuis 1998 gr^ace aux analyses des SNIa.
Ces etudes etant a la frontiere de nombreux domaines dierents de la physique fondamentale (voir l'introduction), je vais essayer de donner une vision claire de la problematique
en detaillant dans l'introduction de ce chapitre les divers elements essentiels. Les articles,
relativement techniques, suivent.
Dans cette introduction, nous rappellerons rapidement les bases theoriques du modele
standard de la cosmologie ce qui nous permettra d'introduire les parametres cosmologiques.
Ensuite nous presenterons les resultats experimentaux les plus marquants qui sont lies aux
etudes qui suivront. Nous detaillerons alors les dierents modeles d'energie noire qui sont
un des ponts entre cosmologie et physique (theorique) des particules. Enn, une analyse
critique de l'extraction des parametres cosmologiques sera donnee. Nous discuterons, en
particulier, le probleme de la degenerescence geometrique et des tres fortes correlations qui
existent entre les divers parametres.
La seconde section est constituee de 4 articles qui traitent de l'extraction des parametres
cosmologiques a partir des SNIa. Les 3 premiers s'interessent aux biais de certains parametres
qui peuvent amener a une interpretation erronee des resultats. Ces biais trouvent leur
origine, du point de vue theorique, dans les fortes degenerescences entre parametres, et du
109
point de vue experimental, dans le grand nombre d'hypotheses qu'il faut faire au niveau de
l'analyse pour extraire les parametres. Nous avons donc entrepris d'etudier les eets de ces
dierentes hypotheses d'analyse. Dans le quatrieme article, nous negligeons les problemes
de biais et nous nous concentrons sur l'optimisation des projets d'observation de SNIa.
Plus precisement, nous essayons de repondre a la question suivante : \Vaut-il mieux un
sondage profond ou un sondage large pour extraire des informations sur la dynamique de
l'energie noire ?" En d'autres termes, nous tentons d'optimiser la profondeur en redshift
ainsi que le nombre total de SNIa a decouvrir, quantites caracterisant le sondage.
An d'eviter le probleme des biais, et an de fournir les meilleures contraintes possibles,
il appara^t indispensable de realiser des analyses de donnees combinant plusieurs sondes
cosmologiques. Cet exercice est relativement dicile car il faut analyser les diverses sondes
dans un m^eme cadre theorique et avec les m^eme hypotheses, ce qui techniquement necessite
la fusion et la comprehension de plusieurs codes numeriques en general assez complexes.
La troisieme partie du chapitre est devouee a ce type d'analyses dites \combinees". Le premier travail a ete realise en 2005 en collaboration avec des equipes du CPPM et de Saclay.
Nous avons combine les donnees les plus recentes (a cette epoque) venant de l'etude des
SNIa et du rayonnement de fond cosmologique (CMB) pour contraindre l'equation d'etat
(dynamique) de l'energie noire. Nous avons realise egalement un travail de prospective
sur les contraintes que l'on pourrait obtenir sur l'equation d'etat en combinant les SNIa
et le CMB avec les mesures de cisaillement gravitationnel (WL). Un second travail a ete
realise, n 2006, en collaboration avec une equipe chinoise de l'institut des hautes energies
de Pekin. Nous avons combine les donnees les plus recentes des SNIa, du CMB mais aussi
des grands sondages de galaxies (spectre de puissance des galaxies mesure par SDSS). Pour
la premiere fois cette analyse tentait de contraindre simultanement l'equation d'etat dynamique de l'energie noire et la courbure de l'Univers.
La derniere partie presente un nouveau test cosmologique a partir des proprietes cinematiques
des galaxies. L'idee originale, essentiellement due a Christian Marinoni, est d'utiliser les
vitesses particulieres des galaxies an de construire des "regles" standard ainsi que des
"chandelles" standard. Le premier article, relativement theorique, presente l'idee de base.
Le second applique cette idee aux mesures faites par la collaboration VVDS qui a realise
un sondage profond de galaxies avec une spectroscopie tres precise.
2.1
Introduction
2.1.1 Le Modele Cosmologique Standard
Les equations de base
Le modele cosmologique standard repose sur la theorie de la relativite generale et sur
le principe cosmologique. Ce dernier stipule que l'on peut considerer l'univers comme
homogene et isotrope aux grandes echelles. L'origine de cette hypothese, du point de vue
110
theorique, remonte a Copernic et au refus de supposer l'existence d'une region (et d'une
direction) particuliere dans l'univers, et a ete rendu populaire par Einstein au debut du
XXeme siecle. Du point de vue observationnel, il est conrme par l'apparente homogeneite
constatee a des echelles superieures a 100 Mpc.
Si on considere un univers maximalement symetrique on est alors amene a utiliser la
metrique de Friedmann-Robertson-Walker (FRW) donnee par :
=
"
#
d2 r
2
2
2
2
+ r (d + sin d )
kr2
+ (t) 1
(2.8)
ou k, l'indice de courbure, est tel que k = 1; 0; 1 correspondant a un univers ouvert,
plat et ferme, respectivement. a(t) est le facteur d'echelle, qui peut ^etre considere comme
\etalon de longueur", et dont la valeur aujourd'hui a0 est normalisee a 1 pour un univers
plat, ou a20 = k=(H02j
T 1j) pour une courbure non-nulle (H0 et T sont des parametres
cosmologiques mesurables dont les denitions suivent). Les variations du facteur d'echelle
sont mesurables gr^ace a la mesure du redshift z qui est une mesure relative de la dilatation
de l'espace entre deux instants, et qui est associe, pour un phenomene lumineux, a la
variation de la longueur d'onde du photon. On a ainsi la relation :
1
a(t) (t)
=
(2.9)
a0
0
1+z
Le parametre de Hubble H (t) decrit l'expansion et est denit par :
a_
H (t) =
(2.10)
ds2
dt2
a2
a
Au temps present on denit H0 = 100 h kms 1Mpc
0:4 < h < 0:9.
avec la contrainte observationnelle
La derivee seconde du facteur d'echelle est relie au parametre de deceleration q(t) :
aa
(2.11)
q(t) = 2
a_
La valeur de ce parametre aujourd'hui, q0, est negative ce qui implique la presence d'une
energie noire lorsque l'on tient compte de la dynamique tiree des equations d'Einstein.
Avant de rentrer dans ces details, continuons la description de la cinematique en introduisant les diverses notions de distances speciques a la cosmologie.
La distance comobile est donnee par :
!
Zz 1
1
0
r(z ) = a0 Sk
dz
(2.12)
a0 0 H (z 0 )
ou Sk (x) = sinh(x); x; sin(x) pour k = 1; 0; 1, respectivement.
Cette distance va servir de base a la denition d'autres distances en cosmologie dont
les interpretations sont beaucoup plus intuitives. La mesure de ces distances est au coeur
de la determination des parametres cosmologiques detaillee dans ce chapitre.
111
1
La distance lumineuse dL(z) relie le ux L emis par un objet de redshift z au ux
observe par unite de surface aujourd'hui :
On peut alors montrer que :
L
4d
2
(2.13)
L
(2.14)
La distance angulaire dA(z) relie la distance propre d'un objet prise perpendiculairement
a la direction radiale (de propagation des photons) l?, a l'angle sous-tendu :
l? dA (2.15)
De cette denition on obtient :
r(z )
2
dA (z ) =
(2.16)
(1 + z) = dL(z)=(1 + z)
dL (z ) = r(z )(1 + z )
A present, nous allons tenir compte de la dynamique decrite par les equations d'Einstein2:
R
(2.17)
2 g = 8G T
Ces equations relient la dynamique de la metrique (et donc du facteur d'echelle a(t)) via
le tenseur de Ricci R et le scalaire de Ricci R, au contenu energetique de l'univers
(represente par le tenseur energie-impulsion T ). En utilisant la metrique FRW et en
supposant que les elements constitutifs de l'univers peuvent ^etre decris par un uide parfait,
nous obtenons les equations de Friedmann :
a_ 2 8G X
k
2
i
(2.18)
=
H =
a
3 i
a2
et
a 4G X
= 3 (i + 3pi)
(2.19)
a
i
ou i et pi sont les densite et pression du uide de nature i.
La premiere equation relie le facteur d'echelle et la courbure aux densites des divers
elements de l'univers. On introduit alors la densite critique c = 8G=3H 2 qui permet
de denir les parametres cosmologiques i = i=c qui sont des rapports de densites. La
premiere equation de Friedmann se simplie alors grandement :
X
T = i = 1 k ;
(2.20)
R
i
ce qui permet de denir le parametre k =
courbure.
k=(a2 H 2 )
pour representer le terme de
2 L'eventuelle presence d'une constante cosmologique sera discutee dans la prochaine section.
112
En general, on considere 3 composantes distinctes, la matiere (i M , qui peut se
decomposer en baryons et matiere noire), la radiation (i R, qui peut se decomposer en
photon et neutrino, i.e. les particules relativistes), et l'energie noire (i X ). Ces trois
composantes ont des evolutions temporelles radicalement dierentes. L'evolution d'un
uide parfait dans un univers en expansion est soumise a l'equation suivante :
_i + 3H (i + pi ) = 0
(2.21)
qui donne la relation integrale suivante :
Zz
i (z )
=
exp
3
(1
+
wi (z 0 )) d ln(1 + z 0 )
(2.22)
i (0)
0
ou wi = pi=i est l'equation d'etat du uide i. On peut alors denir le parametre de Hubble
en fonction des valeurs actuelles des parametres i et de l'equation d'etat de l'energie noire
wX (z ) :
!2
H
(
z
)
(z )
=
(1
+
z )3 m + X X + (1 + z )2 k
(2.23)
E 2 (z ) =
H0
X (0)
On a utilise le fait que la matiere est telle que wM = 0 (impliquant une evolution de la
densite en a 3), et que la radiation correspond a wR = 1=3 (R ' a 4). Ceci implique
que la radiation a domine la dynamique de l'univers avant celle de la matiere. L'equation
d'etat de l'energie noire est inconnue et c'est la quantite physique qui va fortement nous
occuper par la suite. Remarquons seulement que si l'energie noire est associee a une constante cosmologique alors wX = w = 1, la densite est une constante et il appara^t
alors que le destin de l'univers est controle par cette composante inconnue. Cependant,
avant de detailler les divers modeles d'energie noire, nous allons presenter succinctement
les contraintes observationnelles existantes sur ces parametres cosmologiques.
Notons que le modele presente est parfaitement homogene et isotrope, il ne permet
donc qu'une description simpliee de l'Univers, on parle de description du \fond". Si on
veut tenter d'expliquer l'origine des dierentes structures peuplant l'univers, il faut aller
plus loin dans notre description. Les modeles de formation des structures necessitent de
perturber les equations d'Einstein an d'etudier la croissance des uctuations de densite.
Nous verrons que de nombreuses sondes cosmiques sont sensibles a cette physique, et que
l'obtention d'informations ables sur les proprietes de l'energie noire necessitent la prise
en consideration de ces sondes. Cependant, l'ensemble des travaux presentes dans cette
habilitation ne s'interessent principalement qu'a des contraintes venant du fond. Nous ne
detaillerons donc pas cette physique de la croissance des perturbations. Neanmoins, lors de
l'etude des analyses combinees nous inclurons par exemple le CMB, ou l'analyse du spectre
de puissance des galaxies ou encore les eets de cisaillement gravitationnel, qui impliquent
cette physique de la croissance des uctuations primordiales. Cependant l'introduction
a cette physique reste assez lourde et je prefere laisser le lecteur interesse consulter les
references suivantes [37, 38, 39] plut^ot que de surcharger ce manuscrit. Nous allons nous
113
concentrer sur les parametres cosmologiques suivants :
Pour le fond :
les densites i pour les baryons (i = b), la matiere (i = M , incluant baryons et
matiere noire froide), la radiation (i = R), l'energie noire (i = X ) et une eventuelle
courbure spatiale pour l'univers (i = k);
les parametres decrivant une equation d'etat dynamique pour l'energie noire, e.g. w0
et wa (denis dans la section suivante);
la constante de Hubble, e.g. H0 ou h;
un parametre de normalisation du diagramme de hubble, e.g. MS pour les SNIa (qui
inclut H0 mais aussi la magnitude absolue des SNIa, relativement mal connue) .
Pour decrire les inhomogeneites et en particulier les physiques du CMB, du WL et de la
formation des structures on introduit au moins les parametres suivants :
l'indice spectral du spectre de puissance primordial ns;
un parametre de normalisation du spectre de puissance, e.g. 8 ou A selon les
denitions utilisees;
la profondeur optique de reionisation .
Les contraintes observationnelles
Nous discutons a present les mesures qui ont eu le plus d'impact recemment et qui sont en
lien direct avec les etudes des proprietes de l'energie noire.
L'etude des SNIa donne des contraintes fortes sur le phenomene d'acceleration. Les
SNIa sont des objets rares ( 1/siecle/galaxie) mais tres brillants. On les considere comme
des chandelles standard car il appara^t que leurs luminosites (intrinseques) sont comparables. Theoriquement, on le comprend en associant aux SNIa un mecanisme d'explosion
qui change peu d'une SN a l'autre. Plus precisemment, on suppose qu'une SNIa resulte
de l'implosion d'une naine blanche qui vient juste d'atteindre la masse de Chandrasekhar
( 1:4M). On imagine un systeme double ou une etoile, devenue une naine blanche,
accrete la matiere de son etoile compagnon jusqu'a l'instant critique du debut de la supernovae.
Observationnellement, on distingue les SNIa des autres types de SN gr^ace a leurs proprietes photometriques et spectrales. En eet, la courbe de lumiere (evolution de la luminosite dans le temps) des SNIa est caracterisee par une croissance initiale relativement lente,
et le spectre presente une raie d'absorption pour le Silicium a 600 nm. Concretement, les
SNIa peuvent avoir des variations en luminosite tres importantes, mais il existe plusieurs
procedures experimentales dites de \standardisation" qui permettent de reduire fortement
la dispersion de l'echantillon (au maximum de 15% aujourd'hui).
114
HST Discovered
Ground Discovered
45
Binned Gold data
40
ray
-z g
high
35
∆(m-M) (mag)
µ
0.5
n~z
tio
Evolu
pure acceleration: q(z)=-0.5
w=-1.2, d
w/dz=-0
.5
0.0
ΩM =1
.0, Ω
Λ
-0.5
30
t
dus
w=-0.8
=0.0
Empty (Ω=0)
ΩM=0.29, ΩΛ=0.71
0.0
0.5
0.5
, dw/dz
=
~ pure
dece
lerati
on
+0.5
: q(z)=
1.0
z
1.0
z
1.5
1.5
0.5
2.0
2.0
Figure 2.6: Diagramme de Hubble a partir des SNIa les plus recentes [53]. La courbe en
pointillee correspond au best-t en supposant un univers plat caracterise par M = 0:27.
La gure inseree correspond au diagramme de Hubble \residuel" obtenu en soustrayant
aux donnees et a dierents modeles theoriques, un modele d'univers vide [53].
Une fois ces etapes franchies, on peut alors construire un diagramme de Hubble qui
relie la magnitude eective de la SNIa a son redshift. La magnitude apparente eective,
m, est reliee a la magnitude absolue des SNIa, M , et a la distance lumineuse dL par :
dL
!
(2.24)
= 5 log10 Mpc + 25
ou les facteurs numeriques viennent des conventions choisies pour denir m et M en astronomie. Techniquement, pour ne pas trop deteriorer la determination des parametres
cosmologiques, on denit un parametre de normalisation (ou de nuisance) MS qui englobe
les incertitudes sur M et sur H0 :
(2.25)
m(z ) = 5 log10 (DL ) + M 5 log10 (H0 =c) + 25 = MS + 5 log10 (DL )
ou DL(z) (H0=c) dL(z) est la distance lumineuse independante de H0. La gure 2.6
donne le diagramme de Hubble avec les donnees les plus recentes. De ce diagramme on
extrait des contraintes sur les parametres cosmologiques.
Cependant, des que l'on veut extraire des contraintes sur la courbure et/ou sur les proprietes de l'energie noire on est confronte au probleme des degenerescences entre parametres
ce qui nous oblige a contraindre a la main certains parametres (i.e. utilisation de "priors")
ou a realiser des analyses combinees.
=m M
115
Une sonde cosmologique frequemment utilisee est le CMB. La mesure des anisotropies
du fond dius, realisees par COBE en 1992 puis par WMAP depuis 2003, et bient^ot par
Planck, permet de contraindre un grand nombre de parametres associes a des physiques
tres dierentes (expansion/geometrie, ination, reionisation, normalisation du spectre de
puissance des grandes structures ...). En ce qui concerne l'expansion, le CMB fournit
d'excellentes contraintes sur la courbure de l'Univers (parametre T ), mais n'est sensible
qu'a une moyenne de l'equation d'etat de l'energie noire donnee par [40] :
w Z
Z
dz X (z )wX (z )= dz X (z )
(2.26)
An de contraindre wX (z) il est egalement fondamental d'avoir de tres bonnes informations sur M . Les sondages de galaxies sont alors frequemment utilises. Plusieurs types
d'observables de nature tres dierente peuvent ^etre denis comme les comptages d'amas, la
mesure du spectre de puissance des galaxies, les eets de cisaillement gravitationnels (WL)
ou encore la mesure des oscillations baryoniques acoustiques (BAO). Chaque observable a
ses avantages et inconvenients, et en particulier, est plus ou moins sensible aux modeles de
formation des structures et d'energie noire. Nous ne rappellerons pas ici les caracteristiques
et proprietes de chaque type de mesure qui sont bien detaillee dans les revues suivantes
[41, 42, 43].
En combinant ces dierentes sondes cosmologiques on obtient des contraintes plus ou
moins precises. La gure 2.7 donne les contraintes sur X (et la courbure T = M +
X )
lorsque l'on suppose que l'energie noire est une constante cosmologique (i.e. X = ).
ΩΛ
tY
1s
N
LS
S
BAO
1.5
ea
r
No
Bi
g
Ba
ng
2
1
0.5
0
0
g
atin
eler
g
Acc
atin
r
e
l
e
Dec
C
F lo
Op lat sed
en
0.5
ΩM
1
Figure 2.7: Contours dans le plan M -
a partir des observations de SNIa et des BAO.
Les contraintes sont donnees pour chaque sonde separement ou en les combinant, pour des
deges de conance de 68.3%, 95.5% et 99.7% [44].
116
Figure 2.8: Contours dans le plan w-
M a partir des observations de SNIa[45], du CMB[49]
et des BAO[50]. Les contraintes sont donnees pour chaque sonde separement (labels portes
sur la gure, contours a 95%CL) ou en les combinant (traits pleins, contours a 95%CL et
99%CL), tirees de [45].
Si on suppose maintenant que l'energie noire n'est pas forcement une constante cosmologique mais un uide avec une equation d'etat constante (i.e. wX (z) = w = cste), on
obtient alors les contraintes donnees sur la gure 2.8.
Enn, si on veut avoir une description plus generale des modeles d'energie noire, on
doit considerer une equation d'etat dynamique. A cette n, la parametrisation [46, 47]
wX (z ) = w0 + wa z=(1 + z ) est souvent utilisee. On peut alors obtenir des contraintes dans
le plan (w0,wa), ce qui est illustre par la gure 2.9 qui donne les contraintes pour chaque
sonde separement ou en les combinant[45]. Bien qu'il ait ete suppose que l'univers etait
plat, on constate que les contraintes sur l'equation d'etat de l'energie noire sont encore
assez faibles, ce qui nous demande d'avoir une grande ouverture d'esprit lorsque l'on va
considerer les divers modeles d'energie noire.
A partir des 3 gures precedentes on realise que chaque sonde fournit des contraintes
assez "faibles" mais qui indiquent des directions de degenerescence dierentes. Raison
pour laquelle les analyses combinees sont aussi puissantes en fournissant des contraintes
relativement fortes.
Si maintenant on s'interesse aux resultats de ces contraintes on s'apercoit qu'elles tendent vers un modele particulier, appele \modele de concordance", ou CDM , qui est
caracterise par un univers plat avec constante cosmologique tel que M 0:3 et X 0:7.
Une etude plus attentive, en considerant une energie noire avec equation d'etat dynamique,
montre que le modele CDM est au bord des contours a 1. Si on essaye alors de reconstruire l'evolution en redshift de wX (z) on obtient le resultat surprenant que wX (z)
117
Figure 2.9: Contraintes dans le plan (w0,wa) a partir des donnees les plus recentes de
SNIa[45], du CMB[49] et des BAO[50], en supposant que l'univers est plat. Les contraintes
sont donnees pour chaque sonde separement (labels portes sur la gure, contours a 95%CL)
ou en les combinant (traits pleins, contours a 95%CL et 99%CL), tirees de [45].
franchit la barriere problematique w = 1 [52] comme montre sur la gure 2.10.
2.1.2 Modeles d'energie noire
La nouvelle composante \energie noire" doit rendre compte de l'acceleration de l'expansion
(pour des revues consulter [42, 54, 55, 56]). Si on suppose qu'un seul uide est present,
il doit ^etre tel que wX < 1=3 (si on neglige la radiation mais en tenant compte de la
matiere cette relation devient wX < T =(3
X )). On constate donc que l'energie noire
doit avoir une equation d'etat negative, ce qui correspond a un uide relativement exotique.
Le candidat le plus simple est une constante cosmologique ou l'energie du vide, selon
le point de vue adopte. Le modele de concordance, en accord avec les donnees, correspond
a cette hypothese. Cependant, il soure de deux dicultes : le probleme de la constante
cosmologique, fondamental pour les physiciens des particules mais negligeable ou infonde
pour les astronomes, ainsi que le probleme de concidence.
Une constante cosmologique est un terme supplementaire dans les equations d'Einstein
qui deviennent :
R
(2.27)
R
2 g g = 8G T
Historiquement, ce terme a ete introduit par Einstein pour construire un univers statique.
Supprimee par la suite gr^ace a la decouverte de l'expansion et de la notion d'un univers dynamique, elle fut reintroduite recemment pour expliquer l'apparente platitude de l'univers
et l'acceleration recente. Dans les equations de Friedmann on voit appara^tre alors le
118
Figure 2.10: Evolution en redshift de wX (z) a partir des observations les plus recentes
de SNIa[53], du CMB[49] et des BAO[50], tiree de [53]. La courbe pleine est associee a la
parametrisation 2.35, et la courbe en pointillee a un developpement de Taylor du quatrieme
ordre.
terme =3 dans le membre de droite, nous permettant de denir le parametre . Les
contraintes observationnelles
actuelles indiquent 0:7 ce qui se traduit en terme de
densite d'energie par 1=4 10 3 eV .
Pour les astronomes et les specialistes de la relativite generale ce terme est de nature
geometrique et il n'est pas necessaire de lui trouver une origine plus fondamentale. Cependant, pour les physiciens des particules et les experts de theorie quantique des champs, il
appara^t que l'etat fondamental de la theorie, qu'on appelle aussi le vide, est un invariant
de Lorentz. Sa description mathematique est equivalente a la presence d'un uide parfait dont l'equation d'etat vaut wvide = 1. On obtient alors une correspondance entre
cette energie du vide et la constante cosmologique ( = 8Gvide). Pour la cosmologie, la
dierence est surtout technique et il importe peu de savoir si on ajoute dans les equations
d'Einstein un terme a gauche (partie geometrique) ou a droite (partie constitutive). Du
point de vue physique, la dierence est fondamentale, mais la distinction est elle possible?
Aujourd'hui il est dicle de repondre a cette question ...
Le probleme de la constante cosmologique [56, 57, 58] appara^t lorsqu'on essaye d'estimer
la densite d'energie du vide dans le cadre de la theorie quantique des champs. En fait,
le vide soure des m^emes problemes que les particules scalaires a savoir que les corrections radiatives sont quadratiquement divergentes. On a discute ce point dans la section
1.1 a travers le probleme dit hierarchique ou l'on a vu que mH 2cut avec cut le cuto de notre theorie. Pour la densite d'energie du vide on obtient vide 4cut, on parle
aussi de catastrophe ultraviolette. Vu les echelles impliquees en physique des particules
(MEW 100 GeV , MSUSY 1 T eV ou MP 1019 GeV ) on constate que les estimations
1=4
de vide
sont entre 1015 et 1030 trop elevees par rapport a 1=4. Moralite, de grands progres
119
restent a faire pour calculer la densite d'energie du vide en theorie quantique des champs.
Peut-^etre la solution sortira-t-elle des theories quantique de la gravitation, mais nous n'en
sommes pas encore la !
Nous adopterons donc le point de vue de l'astronome par la suite en faisant abstraction
de ce probleme et en parlant de constante cosmologique et non d'energie du vide.
Venons en a present au probleme de concidence qui est a l'origine d'un grand nombre
de modeles d'energie noire : on constate que les densites de matiere (noire) et de constante
cosmologique sont du m^eme ordre de grandeur aujourd'hui ( 0:7c et M 0:3c) alors
que les evolutions temporelles de ces deux composantes sont tres dierentes ( = cste et
M a 3 ). C'est donc une concidence tres surprenante qui a motive la construction d'un
grand nombre de modeles d'energie noire tentant de donner une origine dynamique a ce
phenomene.
Les modeles d'energie noire se decomposent en deux grandes classes. Soit on considere
que nos hypotheses de travail sont justes, et il est alors indispensable de rajouter une nouvelle composante dans l'univers, l'energie noire proprement dite qui est en general decrite
par un nouveau champ scalaire (classique). Soit on considere qu'une de nos hypotheses est
fausse ce qui se traduit par une modication de nos equations de base. Il n'est alors plus
necessaire d'introduire une nouvelle composante, mais la nouvelle dynamique interpretee
dans un cadre standard se traduit par un nouveau uide \eectif", que l'on nomme toujours
energie noire. Nous verrons que la simple distinction entre ces deux classes d'interpretation
demande deja un programme de recherche tres ambitieux.
Modeles avec champ scalaire :
Cette approche est la plus simple car elle conserve nos equations de base. La dynamique de l'energie noire est representee par celle d'un champ scalaire, ce qui necessite
l'introduction d'un potentiel V () de forme inconnue. La description a l'aide d'un lagrangien canonique correspond aux modeles dits de quintessence. An de rendre compte des
contraintes observationnelles qui indiquent une equation d'etat plut^ot exotique (w < 1
recemment), des modeles modiant le lagrangien canonique sont apparus. Nous detaillons
dans la suite ces divers modeles.
Les modeles de quintessence trouvent leurs origines dans les travaux de Ratra-Peebles
[59] et de Wetterich [60], qui reprenaient les idees developpees pour les modeles d'ination
(qui correspond a une acceleration de l'expansion mais dans les instants primordiaux de
l'univers). On suppose que le champ scalaire couple minimalement, c'est a dire qu'il
ressent la gravite a travers la courbure de l'espace-temps et qu'il possede un auto-couplage
represente par le potentiel V (). La dynamique du champ est alors decrite par le lagrangien
canonique suivant :
(2.28)
L = 21 @@ V ():
120
ou le terme cinetique est canonique et le plus simple possible (i.e. lineaire et positif). A
partir du theoreme de Noether on en deduit la forme du tenseur energie impulsion :
p
2
( g L)
T =p
(2.29)
g
g
ou g est le determinant de la metrique.
On peut relier les proprietes du champ scalaire a celles d'un uide parfait, ce qui
simpliera les discussions ulterieures :
1
1
= _ 2 + V () (+ (r)2 )
(2.30)
2
2
1
1
p = _ 2 V () ( (r)2 )
(2.31)
2
6
An d'expliquer une acceleration recente il faut que le champ scalaire roule encore vers
le minimum de son potentiel, ce qui se traduit par une masse du champ extr^emement faible
( 10 30 eV ). La longueur d'onde Compton du champ est alors de l'ordre du rayon de
Hubble ce qui nous laisse supposer que le champ a une distribution spatiale tres douce
dans l'univers observable. On negligera donc le gradient du champ dans les deux equations
precedentes.
On denit alors l'equation d'etat du champ par wX = p= qui est dynamique dans
le cas general i.e. wX wX (z). On constate que si le terme cinetique est faible devant le
potentiel alors la dynamique se rapproche fortement de celle d'une constante cosmologique
(i.e. wX ! 1). Dans tous les cas on peut realiser que cette equation d'etat est bornee
entre 1 et 1 ( 1 wX 1).
L'equation du mouvement du champ scalaire est l'equation de Klein-Gordon :
(2.32)
 + 3H _ = dV=d
qui est equivalente a l'equation de continuite (eq.(2.21)).
Gr^ace a ces formules d'equivalence on peut facilement passer d'une description en terme
de champ scalaire a celle en terme d'un uide parfait. Par la suite nous nous concentrerons
sur ce dernier type de description. Ce choix est renforce [55] par la dicile reconstruction
du potentiel scalaire a partir des donnees. Notons que la description en terme d'un uide
eectif peut englober l'ensemble des modeles d'energie noire.
La dynamique du champ scalaire peut alors se resumer a la connaissance des trois
quantites suivantes [55] : X , wX (z) et wX0 (z) = dwX =d ln a. Les deux dernieres fonctions
permettent de denir un espace des phases pour la dynamique de l'energie noire. A titre
d'exemple, on peut decomposer les modeles de quintessence en deux sous-classes [61] appele
modeles \freezing" ou \thawing", qui dierent par le signe de wX0 comme montre sur la
gure 2.11.
Dierents modeles issus de la physique des hautes energies se placent dans ces deux
sous-classes. Par exemple des modeles avec axions, dilatons, modulis ou encore avec des
121
Figure 2.11: Trajectoires des modeles de quintessence, \freezing" et \thawing", dans
l'espace de phase (wX (z),wX0 (z)), tire de [61].
pseudo bosons de Nambu-Goldstone, rentrent dans la classe \thawing". Les modeles de
supergravite [62] ou des modeles avec brisure dynamique de la supersymetrie, rentrent dans
la classe \freezing". D'autres modeles de nature plus phenomenologique peuvent aussi ^etre
places dans ces sous-classes. Par exemple, les modeles \tracker3" [63] gr^ace a un choix judicieux du potentiel permettent d'obtenir des solutions avec acceleration recente quelque soit
les conditions initiales du champ (ini,0ini), et rentrent dans la classe \freezing". Notons
que les denitions de ces classes de modeles ne sont pas absolues et que certains modeles
de quintessence peuvent ^etre situes en dehors (voir e.g. [64].
Comme une alternative aux modeles de quintessence qui semblent peu en accord avec
les donnees, un autre ensemble de modeles a ete construit. Les modeles de "k-essence" [65]
sont construits a partir d'un lagrangien non canonique (e.g. L = K (@)V ()) ou le r^ole
du terme cinetique est renforce, voir dominant. Ces modeles rentrent, en general, dans la
classe \thawing" [66]. Les modeles "phantom"[67], particulierement exotiques, possedent
la propriete wX < 1 et sont construits a l'aide d'un terme cinetique negatif. Leurs trajectoires dans le plan (wX (z),wX0 (z)) se distinguent aisement de celles des modeles precedents.
Enn, une derniere classe de modeles ou le champ scalaire possede des couplages aux neutrinos [68] semblent interessante. Beaucoup plus de details et de references peuvent ^etre
trouves dans [54].
3 Les modeles de supergravite [62] rentrent aussi dans cette classe.
122
Modeles avec modication des equations de base :
An d'eviter d'inclure directement une nouvelle composante dans le modele cosmologique
standard, on peut changer les equations de Friedmann ou directement les equations d'Einstein.
Dans le premier cas, ce sont les hypotheses d'isotropie et d'homogeneite qui sont remises
en cause. Des modeles tres interessants, dits de "back-reaction" des inhomogeneites [69,
70, 71], relient l'acceleration presente a la formation des structures qui sont le resultat des
inhomogeneites primordiales et de l'instabilite gravitationnelle. Les premiers modeles apparus [69] ont ete fortement critiques [72], en revanche des developpements recents [70, 71]
semblent prometteurs.
De nombreux modeles phenomenologiques [73, 74] ont aussi ete construits dans lesquels
on postule une nouvelle forme pour l'equation de Friedmann, ou la relation entre H 2 et les
densites n'est plus lineaire. Des modeles derives des theories des cordes et des modeles ou
la matiere noire possede des auto-couplages tombent dans cette categorie.
Les extensions de la relativite generale sont nombreuses, et je ne citerai ici que les
modeles suivants ayant obtenus quelques succes :
i) Les theories tenseur-scalaire [75] ou un champ scalaire fondamental couple directement a
la courbure. Ces theories sont tres souvent les manifestations a basse energie des modeles
avec dimensions suplementaires.
ii) Les modeles qui modient l'action d'Hilbert/Einstein par l'ajout de nouveaux termes
de courbure [76, 77].
iii) Les modeles derives des theories des cordes comme le modele DGP [78] ou de Deffayet [79] peuvent predire l'existence de phenomenes tres surprenants comme une autoacceleration.
Plut^ot que de detailler ces divers modeles theoriques, a notre niveau il est plus important
de realiser que ces modeles, interpretes dans un cadre standard (i.e. avec les equations de
Friedmann), sont equivalents a la presence d'un uide aux proprietes non standard. En
eet, on peut denir une equation d'etat eective [55] :
1 d ln H 2
(2.33)
weff 1
3 d ln a
ou H 2 represente la variation du taux d'expansion entre le nouveau modele et le modele
standard sans energie noire :
H 2 (H=H0 )2 M (1 + z )3 ( k (1 + z )2 )
(2.34)
Par consequent, il est aussi possible de denir des trajectoires dans le plan (wX (z),wX0 (z))
ou wX (z) et wX0 (z) sont a present eectifs. Par exemple, la gure 2.12 donne quelques trajectoires de modeles phenomenologiques modiant les equations de Friedmann de la facon
123
suivante H 2 H [74]. Les modeles avec = 0:5 et = 1 sont representes [55]. Remarquons que ce dernier modele correspond au modele DGP [78] derive des theories des cordes.
Figure 2.12: Trajectoires des modeles du type H 2 H avec = 0:5 et = 1 (equivalent
au modele DGP), dans l'espace de phase (wX (z),wX0 (z)), tire de [55]. Les lignes pointillees
delimitent la zone de la classe \freezing". La courbe du modele DGP est continue jusqu'a
z = 0, les croix indiquent les valeurs a z = 1; 2; 3.
Les problemes :
Utiliser une approche basee sur la determination de l'equation d'etat pour distinguer
les divers modeles d'energie noire est pertinent car cela permet de denir une methode
independante de tout modele specique. Cependant les problemes sont nombreux dont
voici une liste minimale :
* Necessite de denir une parametrisation :
La caracterisation de l'energie noire passe, au moins, par la connaissance d'un nombre (
X )
et de deux fonctions wX (z) et wX0 (z). Malheureusement, les donnees observationnelles ne
peuvent pas contraindre un nombre eleve de parametres. Des etudes recentes montrent que
l'on peut esperer contraindre au plus deux parametres caracterisant wX (z) [80]. Avec les
analyses combinees ce chire peut sans doute augmenter un peu mais cela reste a etudier.
Il faudrait donc avoir une intuition physique pour imaginer grossierement la forme de ces
124
fonctions, ce qui est tres audacieux aujourd'hui vu le grand nombre de modeles proposes.
Cependant, un consensus se forme autour de l'utilisation de la parametrisation suivante
[46, 47]:
z
(2.35)
wX (z ) = w0 + wa
1+z
qui semble posseder des proprietes interessantes et en particulier reproduit correctement
bon nombre de modeles (voir [55] pour plus de details). En particulier, w0 est la valeur de
l'equation d'etat aujourd'hui, et wa est une mesure de la variation temporelle wX0 :
wa ( wX0 =a)jz=1 = 2wX0 (z = 1):
(2.36)
Bien-entendu, cette description a deux parametres ne peut pas decrire toutes les dynamiques possibles. Par exemple, la forme donnee par l'eq.(2.35) impose une relation
lineaire dans l'espace de phase (wX (z),wX0 (z)), car on a la relation wX0 = wX w1 ou
w1 = wX (z ! 1) = w0 + wa .
D'autres parametrisations ont ete proposees, voir [81] pour une liste recente, mais
aucune ne semble plus adaptee que la parametrisation de l'eq.(2.35). Par exemple, un
simple developpement de Taylor en z a tout d'abord ete propose : wX (z) = w0 + w1z, mais
des que l'on tiend compte du CMB on a besoin de calculer wX a tres haut redshift (i.e.
zCMB 1089) et on s'apercoit que wX devient tres grand (si w1 > 0) ce qui fait diverger
la densite d'energie noire. Il est donc indispensable de faire attention au comportement a
haut z de toute parametrisation. En particulier, le CMB impose w1 = wX (z ! 1) < 0
[82].
Il est aussi possible de denir des approches n'utilisant aucune parametrisation, comme
l'analyse en composantes principales [83], la \derivation" des donnees [84, 85] ou encore
des techniques par iteration adaptative [86]. Il appara^t que ces methodes ont leurs propres
avantages et inconvenients, ce qui ne les rend pas competitives par rapport a l'approche
parametrique mais plut^ot complementaires.
* Degenerescence des parametres cosmologiques :
Une fois une parametrisation de wX (z) choisie on peut denir l'ensemble des parametres
cosmologiques intervenant dans le parametre de Hubble H (z) (voir eq.(2.23)), a savoir M ,
X , w0 et wa. Les distances, qui sont mesurees experimentalement, sont relies au niveau
de la theorie au parametre de Hubble H (z) a travers une relation integrale (eq.(2.14) et
eq.(2.12)). Cette relation integrale va produire de tres fortes degenerescences entre les
divers parametres [87, 88], ce qui rend leur determination dicile. Nous allons detailler
ce point dans la prochaine section car il est a l'origine de l'existence de nombreux biais
d'analyse ce qui motive la realisation des analyses combinees an de rendre moins severe
ce probleme.
* Necessite de tenir compte de la croissance des structures et/ou d'autres tests :
Il appara^t que plusieurs modeles d'energie noire issus de classes dierentes peuvent correspondre exactement aux m^emes equations d'etat eectives. Par exemple, les theories
125
tenseur-scalaire sont equivalentes aux modeles avec termes de courbure supplementaires
[76]. De m^eme certains modeles de backreactions sont equivalents aux modeles de quintessence [70]. L'equation d'etat et sa derivee ne sont donc plus susant pour caracteriser
les modeles d'energie noire. Il faut utiliser d'autres observables qui vont aller plus loin
dans la distinction entre modeles. De grands espoirs se fondent sur les contraintes que
l'on peut obtenir a travers l'etude de la croissance des structures. Par exemple, Linder
a montre recemment [89] qu'une parametrisation du facteur de croissance pourrait ^etre
utile a la distinction des modeles respectant ou non la relativite generale. Pour certains
modeles, en particulier ceux issus de la physique des hautes energies, il sera necessaire
de tenir compte de nombreux tests dierents, comme les resultats du LHC sur l'existence
de nouvelles particules, forces ou symetries, les tests locaux de la relativite generale, la
verication de la relation de dualite entre distances (i.e. relation entre dL(z) et dA(z), voir
eq.(2.16)), le test de l'equation de Poisson ou encore le test du principe d'equivalence ....
Cette problematique est relativement bien decrite dans [90, 91].
2.1.3 La degenerescence geometrique
Initialement la degenerescence geometrique a ete traitee dans le cadre de l'etude des
anisotropies du CMB, mais en fait elle est generique a l'utilisation de distances (lumineuses
ou angulaires) en cosmologie. Le cur duR probleme vient de la relation integrale entre les
distances et le parametre de Hubble d 1=H (z) (voir les equations (2.14) et (2.16)).
La degenerescence geometrique est telle que l'on peut avoir des spectres de puissance
des anisotropies du CMB identiques pour des evolutions du bruit de fond tres dierentes
[92, 93, 87, 40]. Pour cela il faut satisfaire trois conditions :
* contenus en baryons et en matiere noire identiques,
* spectre de uctuations primordiales identiques,
* parametres de decalage R (\CMB shift parameter") identiques.
Le parametre R est deni par : p
!
Z zCMB 1
q
M
0
dz
(2.37)
R = q Sk j
k j
E (z 0 )
0
j
k j
q
= (1 + z)H0 M dA(zCMB )
(2.38)
ou zCMB = 1089 [49], Sk (x) = sinh(x); x; sin(x) pour k > 0 (k = 1; T < 1); k =
0 (k = 0; T = 1); k < 0 (k = 1; T > 1) respectivement, et
!
H (z ) 2
(z )
2
E (z ) =
=
(1
+
z )3 m + X X + (1 + z )2 k ;
(2.39)
H0
X (0)
ou XX ((0)z) est donne par l'eq.(2.22) qui avec la parametrisation donnee par l'eq.(2.35) fournit:
X (z )
= (1 + z)3(1+w +wa)e 3waz=(1+z):
(2.40)
X (0)
126
0
Figure 2.13: Haut : Spectre de puissance du CMB pour les modeles representes par les 5
points de la gure 2.14. Bas : eets residuels de chaque modele par rapport a un univers
plat avec M = 1, les valeurs pres de chaque courbe donnent h2. Les deux lignes externes
hachurees donnent les deviations standard duent a la variance cosmique, indiquant donc
que les divers modeles sont indistinguables. Figure tiree de [87].
Dans le cadre de la theorie lineaire des perturbations, aucune mesure des anisotropies du
CMB, quelque soit leur precision, ne permet de briser cette degenerescence[87]. Cela impose
des limites fondamentales sur la reconstruction de la courbure, du parametre de Hubble
H0 et sur les proprietes de l'energie noire [87, 40]. Pour illustrer ceci la gure 2.13 donne le
spectre de puissance des anisotropies du CMB pour des modeles dierents representes par
les 5 points de la gure 2.14 mais veriant les 3 conditions de degenerescence geometrique.
Ces modeles dierents uniquement par leurs valeurs de k et mais possedent la m^eme
valeur de R ce qui est illustre sur la gure 2.14. Ces modeles sont donc degeneres du
point de vue du CMB. La gure 2.14 donne les degenerescences pour plusieurs valeurs de
R, montrant ainsi que pour des valeurs de R raisonnables on a une innite de modeles
degeneres. Cette degenerescence entre la courbure et une constante cosmologique, i.e.
entre k et , a ete etudiee en details dans [87] ou il est montre que sans information
exterieure les erreurs sur k et sont de l'ordre de 0.1 et 1, respectivement, et ce quelque
soit la precision des mesures du CMB.
Si on elargit notre espace des parametres pour aller au-dela d'une constante cosmologique, comme par exemple en supposant des modeles d'energie noire avec wX = cste
127
Figure 2.14: Lignes de degenerescence avec R constant dans le plan (
k ,
). Les valeurs de
R sont donnees pres de chaque courbe. Les 5 points donnent les positions de chaque modele
ayant un spectre de puissance indistinguable des autres. Le poin a l'origine correspond au
modele SCDM tel que T = M = 1. Figure tiree de [87].
ou avec une equation d'etat dynamique, on renforce ce probleme de degenerescence. On
pourra consulter [40] qui donne les lignes de degenerescence dans le plan (
M ,w) pour
un univers plat. La gure 2.15 donne les modeles degeneres dans le plan (w0,wa) pour
dierentes valeurs des parametres de densite.
Chaque point du plan (w0,wa) correspond a un modele particulier pour lequel on calcule
R. An de voir quels modeles sont degeneres, les contours sont obtenus en demandant
R = 1:70 0:03, qui est la valeur actuelle[94] deduite des dernieres mesures de WMAP[49].
On peut remarquer que cette valeur est obtenue sous de nombreuses hypotheses (e.g.
univers plat avec constante cosmologique) et que cela peut avoir un fort impact sur le
resultat [95], mais cela importe peu pour nos propos.
A partir de la gure 2.15, on constate que la position et la taille de la zone degeneree
varient beaucoup avec les parametres de densite :
Pour les modeles ouverts (plot en haut a gauche avec M = 0:3 et X = 0:6) la zone
degeneree se reduit a une ligne tres ne.
Pour les modeles plats (plot en haut a droite avec M = 0:3 et X = 0:7) on obtient
128
Figure 2.15: Degenerescence geometrique dans le plan (w0,wa) pour M = 0:3 et X = 0:6
(en haut a gauche); M = 0:3 et X = 0:7 (en haut a droite); M = 0:3 et X = 0:74
(en bas a gauche); M = 0:3 et X = 0:8 (en bas a droite). Les contours correspondent a
R = 1:70 0:03, est la limite CMB w1 = w0 + wa > 0 est donnee.
une ligne tres ne pour les modeles phantom mais la zone degeneree augmente dans
la partie w0 > 1.
Pour les modeles legerement fermes tels que 1 < T 1:06 (plot en bas a gauche avec
M = 0:3 et X = 0:74), on obtient une zone degeneree extr^emement importante. La
degenerescence geometrique est donc particulierement problematique et les mesures
du CMB ne peuvent pas distinguer ces divers modeles.
Pour les modeles fermes avec T > 1:06 (plot en bas a droite avec M = 0:3 et
X = 0:8), R est bien en-dessous de la valeur 1.70 pour toutes valeurs de (w0,wa). Par
consequent, la degenerescence geometrique n'est pas un probleme pour les modeles
fermes avec T > 1:06, sauf si les contraintes du CMB sur R reduisent fortement la
valeur centrale obtenue (par exemple a cause de nouvelles hypotheses d'analyse).
Ces resultats permettent de comprendre indirectement les resultats experimentaux
actuels, en particulier ceux contraignant simultanement la courbure et une equation d'etat
dynamique pour l'energie noire presentes en section 3 de ce chapitre, qui indiquent une
preference pour les modeles legerement fermes mais absolument compatible avec les modeles
plats. C'est pour cet ensemble de parametres que l'espace de phase est le plus grand, et
de tres loin. Les modeles fermes avec T > 1:06 sont exclus car incapables de fournir R =
1:70 0:03 quelque soit (w0,wa). Pour les modeles ouverts la degenerescence geometrique
129
est operationnelle : on peut avoir R = 1:70 0:03 avec M 0:3 pour satisfaire les contraintes issus des grandes structures et avec des couples (w0,wa) en accord avec les mesures
de SNIa. Cependant ces modeles ouverts sont exclus, ou plut^ot negliges, par les methodes
statistiques habituelles, car l'espace de phase associe est extr^emement faible.
Les problemes rencontres precedemment se retrouvent pour toutes les autres sondes
cosmologiques avec quelques nuances. Par exemple, pour les SNIa les degenerescences
entre les parametres sont dues[88] a la forme de la distance lumineuse (eq.(2.14)) mais
a present la borne d'integration n'est plus zCMB = 1089 mais le redshift de la SNIa la
plus lointaine qui aujourd'hui est zSN = 1:7. Cette dierence a son importance car elle
va permettre de briser partiellement les degenerescences lors de la combinaison des deux
sondes. La brisure n'est que partielle car malheureusement la description des anisotropies
du CMB implique tellement de parametres qu'il y a de nombreuses degenerescences en sus
de la geometrique. On pourra consulter [82, 96] qui discutent en details ces problemes pour
le CMB. Les travaux presentes dans la prochaine section traitent ces problemes dans le cas
des SNIa.
2.2 Supernovae : biais et prospectives
Dans cette partie nous nous concentrons sur l'etude des SNIa. Les parametres cosmologiques
concernes sont au nombre de cinq : M , X , w0, wa (ou w14) et MS le parametre de normalisation des SNIa. Ce dernier parametre est surtout contraint par les SNIa proches (i.e.
z < 0:1) et nous l'etudierons tres peu. Il appara^t que les SNIa seules ne peuvent pas
contraindre les quatre parametres restant. Si on essaye de le faire on obtient des erreurs
tres grandes, m^eme avec les projets de mesure de plus de 2000 SNIa (e.g. SNAP). On est
donc oblige de contraindre a la main (ou gr^ace a des analyses combinees, voir la prochaine
section) deux de ces parametres. Les hypotheses qui sont generalement faites sont les
suivantes :
On suppose que l'energie noire est une constante cosmologique (wX = 1). Les quantites que l'on cherche a contraindre sont alors M et X , et donc la courbure. Un
article est en cours de redaction sur ce type d'approche mais dans le cadre d'analyses
combinees.
On suppose que l'univers est plat, ce qui impose la relation X = 1 M et donc
supprime un parametre. La deuxieme hypothese concerne l'equation d'etat qui est
choisie constante wX (z) = w = cste. Le premier article s'interesse aux eets de cette
derniere hypothese qui peuvent ^etre tres important, surtout sur la determination de
M .
4 Pour
les deux premiers articles nous avons utilis
e le d
eveloppement de Taylor au premier ordre de
l'
equation d'
etat :
wX (z )
w0 + w1 z .
w1 wa =2.
=
a grossi
erement la relation
Pour les SN seules, les r
esultats avec
130
wa
sont tr
es similaires et on
On suppose que l'univers est plat et on rajoute une contrainte exterieure sur M
(prior). Si l'erreur supposee pour M est petite, cela revient a eliminer du jeu les
densites et les parametres restants sont w0 et wa. Le deuxieme article etudie les eets
d'un prior errone sur M . Nous verrons que le choix M = 0:27 0:04 utilise par
plusieurs collaborations experimentales force les donnees a converger vers le modele
CDM quelque soit la veritable cosmologie.
Le troisieme article etudie les contraintes directes sur l'acceleration. En eet, il est
possible de relier directement la distance lumineuse au parametre de deceleration q(z).
Avec cette approche on n'utilise pas les equations de Friedmann mais on a besoin d'une
prescription pour decrire q(z). La methode utilisee par Riess et collaborateurs [97], etait
de supposer une evolution lineaire (i.e. q(z) = q0 + q1z). Nous avons montre que cette
hypothese lineaire est dangereuse pour l'interpretation des resultats. Depuis, les analyses
tentant de contraindre q(z) directement sont plus ranees [98].
Dans le quatrieme article, nous negligeons les problemes de biais et nous nous concentrons sur l'optimisation des projets d'observation de SNIa. Plus precisement, nous essayons
de repondre a la question suivante : \Vaut-il mieux un sondage profond ou un sondage
large pour extraire des informations sur la dynamique de l'energie noire ?" En d'autres
termes, nous tentons d'optimiser la profondeur en redshift ainsi que le nombre total de
SNIa a decouvrir caracterisant le sondage.
2.2.1 Biais d^
u a une equation d'etat dynamique
Article publie sous la reference : Phys. Rev. D70 (2004) 043514.
131
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2.2.3 Biais d^
u a une evolution non-lineaire de q (z )
Article publie sous la reference : Phys. Rev. D72 (2005) 061302.
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2.2.4 Prospective SNIa : champ large ou profond ?
Article publie sous la reference : Astronomy and Astrophysics 464 (2007) 837.
176
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2.3 Analyses combinees
An d'eviter le probleme des biais, et dans le but de fournir les meilleures contraintes
possibles, il appara^t indispensable de realiser des analyses de donnees combinant plusieurs
sondes cosmologiques. Cependant, les sondes faisant appel aux modeles de uctuations
primordiales de densite soit directement (e.g. CMB) soit indirectement a travers la seule
utilisation du spectre de puissance aujourd'hui (e.g. grandes structures), font intervenir
beaucoup plus de parametres cosmologiques (et astrophysiques) que les seules etudes de
distances comme les SNIa. Cette multitude de parametres nous force tout de m^eme a
eectuer un certain nombre d'hypotheses. Par ailleurs, chaque sonde a ses propres sources
d'erreurs systematiques qui trouvent leurs origines dans les incertitudes de mesure ou
de calibration de nature purement astrophysique mais aussi dans les aspects purement
experimentaux des appareillages fournissant les mesures.
Dans les articles qui vont suivre nous avons neglige ces problemes de biais potentiels et
d'erreurs systematiques. Nous avons surtout cherche a obtenir des contraintes directement
a partir des donnees. Nous presentons egalement quelques travaux de prospectives realises
an d'estimer la puissance des analyses combinees.
Le premier travail a ete realise en 2005 en collaboration avec des equipes du CPPM
et de Saclay. Nous avons combine les donnees les plus recentes (a cette epoque) venant
de l'etude des SNIa et du rayonnement de fond cosmologique pour contraindre l'equation
d'etat (dynamique) de l'energie noire. Nous avons egalement realise un travail de prospective sur les contraintes que l'on pourrait placer sur cette equation d'etat en combinant les
SNIa et le CMB avec les mesures de cisaillement gravitationnel.
Un second travail a ete realise, n 2006, en collaboration avec une equipe chinoise de
l'institut des hautes energies de Pekin. Nous avons combine les donnees les plus recentes
des SNIa, du CMB, mais aussi des grands sondages de galaxies (spectre de puissance des
galaxies mesure par SDSS). Pour la premiere fois cette analyse tentait de contraindre simultanement l'equation d'etat dynamique de l'energie noire ainsi que la courbure de l'Univers.
Le programme d'etude futur sera justement d'estimer l'impact des biais et des erreurs systematiques quelle que soit leur nature, sur la determination des parametres cosmologiques. Comme mentionne precedemment un travail est en cours sur les eets de
l'hypothese wX = 1 sur la reconstruction de la courbure dans le cadre d'une nalyse
combinant les SNIa, le CMB et les BAO.
2.3.1 Combinaison SN+CMB et prospective avec WL sur les
proprietes de l'energie noire
Article publie sous la reference : Astronomy and Astrophysics 448 (2006) 831.
192
Prospects for Dark Energy Evolution:
a Frequentist Multi-Probe Approach
Ch. Yèche1 , A. Ealet2 , A. Réfrégier1 , C. Tao2 , A. Tilquin2 , J.-M. Virey3 , and D. Yvon1
1
2
3
DSM/DAPNIA, CEA/Saclay, F-91191, Gif-sur-Yvette, France,
Centre de Physique des Particules de Marseille, CNRS/IN2P3-Luminy and Universit´e de la M´editerran´ee, Case 907, F-13288
Marseille Cedex 9, France,
Centre de Physique Th´eorique? , CNRS-Luminy and Universit´e de Provence, Case 907, F-13288 Marseille Cedex 9, France.
September 9, 2007
Abstract. A major quest in cosmology is the understanding of the nature of dark energy. It is now well known that a combination
of cosmological probes is required to break the underlying degeneracies on cosmological parameters. In this paper, we present a
method, based on a frequentist approach, to combine probes without any prior constraints, taking full account of the correlations
in the parameters. As an application, a combination of current SNIa and CMB data with an evolving dark energy component
is first compared to other analyses. We emphasise the consequences of the implementation of the dark energy perturbations
on the result for a time varying equation of state. The impact of future weak lensing surveys on the measurement of dark
energy evolution is then studied in combination with future measurements of the cosmic microwave background and type Ia
supernovae. We present the combined results for future mid-term and long-term surveys and confirm that the combination with
weak lensing is very powerful in breaking parameter degeneracies. A second generation of experiment is however required to
achieve a 0.1 error on the parameters describing the evolution of dark energy.
Key words. cosmology: cosmological parameters – supernovae – CMB – gravitational lensing – large-scale structure in the
universe – dark energy – equation of state – evolution
1. Introduction
Supernovae type Ia (SNIa) observations (Knop et al. 2003, Riess et al. 2004) provide strong evidence that the universe
is accelerating, in very good agreement with the WMAP Cosmic Microwave Background (CMB) results (Bennett et al. 2003,
Spergel et al. 2003) combined with measurements of large scale structures (Hawkins et al. 2003, Tegmark et al. 2004). The
simplest way to explain the present acceleration is to introduce a cosmological constant in Einstein’s equations. Combined with
the presence of Cold Dark Matter, it forms the so-called ΛCDM model. Even if this solution agrees well with current data, the
measured value of the cosmological constant is very small compared to particle physics expectations of vacuum energy, requiring
a difficult fine tuning. A favourite solution to this problem involves the introduction of a new component, called ”dark energy”
(DE), which can be a scalar field as in quintessence models (Wetterich 1988, Peebles & Ratra 1988).
The most common way to study this component is to measure its ”equation of state” (EOS) parameter, defined as w = p/ρ ,
where p is the pressure and ρ the energy density of the dark energy. Most models predict an evolving equation w(z). It has been
shown (e.g., Maor et al. 2001, Maor et al. 2002, Virey et al. 2004a, Gerke & Efstathiou 2002) that neglecting such evolution
biases the discrimination between ΛCDM and other models. The analysis of dark energy properties needs to take time evolution
(or redshift z dependence) into account.
Other attractive solutions to the cosmological constant problem imply a modification of gravity (for a review, cf., e.g., Lue
et al. 2004, or Carroll et al. 2005 and references therein). In this case, there is no dark energy as such and thus no dark energy
equation of state. In this paper, we consider only the dark energy solution, keeping in mind that Lue et al. (2004), among others,
have shown that the induced changes in the Friedmann equations could be parameterised in ways very similar to a dark energy
evolving solution.
?
“Centre de Physique Th´eorique” is UMR 6207 - “Unit´e Mixte de Recherche” of CNRS and of the Universities “de Provence”, “de la
M´editerran´ee” and “du Sud Toulon-Var”- Laboratory affiliated to FRUMAM (FR 2291).
2
Ch. Yèche et al.: Prospects for Dark Energy Evolution
As various authors have noted (e.g., Huterer & Turner 2001, Weller & Albrecht 2002), SNIa observations alone will not be
able to distinguish between an evolving equation of state and ΛCDM. This technique indeed requires prior knowledge of the
values of some parameters. In particular, the precision on the prior matter density Ω m has an impact on the constraints on the
time evolution of the equation of state w, even in the simplest flat Universe cosmology (e.g., Virey et al. 2004b).
Extracting dark energy properties thus requires a combined analysis of complementary data sets. This can be done by combining SNIa data with other probes such as the CMB, the large scale distributions of galaxies, Lyman α forest data, and, in the
near future, the observation of large scale structure with the Sunyaev-Zeldovich effect (SZ) (Sunyaev & Zeldovich 1980) or with
weak gravitational lensing surveys (WL), which provide an unique method to directly map the distribution of dark matter in the
universe (for reviews, cf., e.g., Bartelmann & Schneider 2001, Mellier et al. 2002; Hoekstra et al. 2002, Refregier 2003, Heymans
et al. 2005 and references therein).
Many combinations have already been performed with different types of data and procedures, (e.g., Bridle et al. 2003, Wang
& Tegmark 2004, Tegmark et al. 2004, Upadhye et al. 2004, Ishak 2005, Seljak et al. 2004, Corasaniti et al. 2004, Xia et al.
2004). All studies have shown the consistency of existing data sets with the ΛCDM model and the complementarity of the
different data sets in breaking degeneracies and constraining dark energy for future experiments. But the results differ by as
much as 2σ on the central values of the parameters describing an evolving equation of state.
In this paper, we have chosen three probes, which seem to best constrain the parameters of an evolving equation of state
when combined, namely, SNIa, CMB and weak lensing. Considering a flat Universe, we combine the data in a coherent way,
that is to say, under identical assumptions for the dark energy properties for the three probes, and we completely avoid the use
of priors. This had not always been done systematically in all previous combinations. We also adopt a frequentist approach for
the data combination, where the full correlations between the cosmological parameters are taken into account. This method
allows us to provide, simultaneously, confidence intervals on a large number of distinct cosmological parameters. Moreover, this
approach is very flexible as it is easy to add or remove parameters in contrast with other methods.
The paper is organised as follows: In Sec. 2, we describe our framework and statistical procedure, based on a frequentist approach, which can accommodate all parameters without marginalisation. For our simulation and analysis, we use the CMBEASY
package for CMB (Doran 2003), the Kosmoshow program for SNIa (Tilquin 2003) and an extension of the calculations from
Refregier et al. (2003) for weak lensing. In each case, the programs take into account the time evolution of the equation of state
(cf Sec. 2.2 for details).
In Sec. 3, we apply this method to current data sets of SNIa and WMAP data. We first verify that the constraints on the
cosmological parameters estimated with a Fisher matrix technique (Fisher 1935), are consistent with those obtained with a
complete error analysis. We then compare these errors with other works and discuss the differences. In particular, we discuss how
the treatment of the dark energy perturbations can explain some of the differences found in the literature.
In Sec. 4, we study the statistical sensitivities of different combinations of future surveys. We simulate expectations for the
ground surveys from the Canadian French Hawaii Telescope Legacy Surveys (CFHTLS) and new CMB data from Olimpo as
well as the longer term Planck and SNAP space missions. For these future experiments, the results are combined with a Fisher
matrix technique, compared and discussed.
Finally, our conclusions are summarised in Sec. 5.
2. Combination method
In this section, we first summarise the framework used in this paper, and describe our approach based on frequentist statistics.
2.1. Dark Energy Parametrization
The evolution of the expansion parameter is given by the Hubble parameter H through the Friedmann equation
!2
H(z)
ρX (z)
ΩX + (1 + z)2 Ωk ,
= (1 + z)3 Ωm +
H0
ρX (0)
(1)
with
#
" Z z
ρX (z)
1 + w(z0 ) d ln(1 + z0 )
= exp 3
ρX (0)
0
(2)
where the ratio of the dark energy density to the critical density is denoted Ω X in a general model and ΩΛ in the simplest case
of a Cosmological Constant (w = −1). Ω M is the corresponding parameter for (baryonic+cold dark) matter. Note that we have
neglected the radiation component ΩR . The present total and curvature density parameters are Ω and Ω κ = 1 − Ω, respectively.
The present value of the Hubble constant is parameterised as H0 = 100h km s−1 Mpc−1 .
Ch. Yèche et al.: Prospects for Dark Energy Evolution
3
As it is not possible to constrain a completely unknown functional form w(z) of the time evolution of the equation of state,
we adopt a parametric representation of the z dependence of the equation of state. We need this parametric form to fit all the data
sets over a large range of z: from z ' 0 − 1 for the SNIa and weak lensing, up to z ' 1100 for the CMB. For this purpose, we
choose the parametrization proposed by Chevallier & Polarski (2001) and Linder (2003) :
w(z) = w0 + wa z/(1 + z),
(3)
which has an adequate asymptotic behaviour. In this paper, we thus use two parameters, w 0 and wa , to describe the time evolution
of the equation of state (see justifications in Linder & Huterer 2005). For this parametrization of w(z), Eq. 2 reduces to:
ρX (z) = ρX (0) e−3waz/(1+z) (1 + z)3(1+w0 +wa ) .
For a constant w ≡ w0 (wa = 0), the usual form ρX (z) = ρX (0) (1 + z)3(1+w0 ) is recovered.
The comoving distance χ is defined as
Z z
c
χ(z) =
dz0 ,
0
0 H(z )
(4)
(5)
and the comoving angular-diameter distance r(χ) is equal, respectively, to χ, R 0 sin(χ/R0 ), R0 sinh(χ/R0 ), for a flat, closed and
open Universe where the present curvature radius of the universe is defined as R0 = c/(κH0 ) with respectively κ2 ≡ 1, −Ωκ , and
Ωκ .
2.2. Statistical approach
Most recent CMB analysis use Markov Chains Monte Carlo simulations (Gilks et al. 1996, Christensen & Meyer 1998) with
bayesian inference. The philosophical debate between the bayesian and the frequentist statistical approaches is beyond the scope
of this paper (for a comparison of the two approaches see, for instance, Feldman & Cousins 1998 and Zech 2002). Here, we
briefly review the principles of each approach.
For a given data set, the bayesian approach computes the probability distribution function (PDF) of the parameters describing
the cosmological model. The bayesian probability is a measure of the plausibility of an event, given incomplete knowledge. In
a second step, the bayesian constructs a ’credible’ interval, centered near the sample mean, tempered by ’prior’ assumptions
concerning the mean. On the other hand, the frequentist determines the probability distribution of the data as a function of the
cosmological parameters and gives a confidence level that the given interval contains the parameter. In this way, the frequentist
completely avoids the concept of a PDF defined for each parameter. As the questions asked by the two approaches are different,
we might expect different confidence intervals. However, the philosophical difference between the two methods should not
generally lead, in the end, to major differences in the determination of physical parameters and their confidence intervals when
the parameters stay in a physical region.
Our work is based on the ’frequentist’ (or ’classical’) confidence level method originally defined by Neyman (1937). This
choice avoids any potential bias due to the choice of priors. In addition, we have also found ways to improve the calculation
speed, which gives our program some advantages over other bayesian programs. Among earlier combination studies (e.g., Bridle
et al. 2003, Wang & Tegmark 2004, Tegmark et al. 2004, Upadhye et al. 2004, Ishak 2005, Seljak et al. 2004, Corasaniti et
al. 2004, Xia et al. 2004) only that of Upadhye et al. (2004) uses also a frequentist approach.
2.2.1. Confidence levels with a frequentist approach
For a given cosmological model defined by the n cosmological parameters θ = (θ1 , . . . , θn ), and for a data set of N quantities
x = (x1 , . . . , xN ) measured with gaussian experimental errors σ x = (σ1 , . . . , σN ), the likelihood function can be written as:
!
(xi − xi,model )2
1
.
(6)
L(x, σ x ; θ) = √
exp −
2σ2i
2πσi
where xmodel = (x1,model , . . . , xN,model ) is a set of corresponding model dependent values.
In the rest of this paper, we adopt a χ2 notation, which means that the following quantity is minimised:
χ2 (x, σ x ; θ) = −2 ln(L(x, σ x ; θ))
(7)
We first determine the minimum χ20 of χ2 (x, σ x ; θ) letting free all the cosmological parameters. Then, to set a confidence level (CL)
on any individual cosmological parameter θi , we scan the variable θi : for each fixed value of θi , we minimise again χ2 (x, σ x ; θ)
but with n − 1 free parameters. The χ2 difference, ∆χ2 (θi ), between the new minimum and χ20 , allows us to compute the CL on
the variable, assuming that the experimental errors are gaussian,
4
Ch. Yèche et al.: Prospects for Dark Energy Evolution
1 − CL(θi ) = √
1
2Ndo f Γ(Ndo f /2)
Z
∞
e−t/2 tNdo f /2−1 dt
(8)
∆χ2 (θi )
where Γ is the gamma function and the number of degrees of freedom N do f is equal to 1. This method can be easily extended
to two variables. In this case, the minimisations are performed for n − 2 free parameters and the confidence level CL(θi , θ j ) is
derived from Eq. 8 with Ndo f = 2.
By definition, this frequentist approach does not require any marginalisation to determine the sensitivity on a single individual
cosmological parameter. Moreover, in contrast with bayesian treatment, no prior on the cosmological parameters is needed. With
this approach, the correlations between the variables are naturally taken into account and the minimisation fit can explore the
whole phase space of the cosmological parameters.
In this study, the minimisations of χ2 (x, σ x ; θ) are performed with the MINUIT package (James 1978). For the 9 parameter
study proposed in this paper, each fit requires around 200 calculations of χ2 . The consumed CPU-time is dominated by the
computation of the angular power spectrum (C ` ) of the CMB in CMBEASY (Doran 2003). In practice, to get the CL for one
variable, as shown, for instance, in Fig. 1, the computation of the C ` is done around 10000 times. The total number of calls to
perform the study presented in Tab. 1, is typically 3 or 4 times smaller than the number of calls in the MCMC technique used by
Tegmark et al. (2004). This method is very powerful for studying the impacts of the parameters: it is not costly to add or remove
parameters because the number of C ` computations scales with the number of parameters, in contrast with the MCMC method,
which requires the generation of a new chain.
2.2.2. Combination of cosmological probes with Fisher matrices
In parallel with this frequentist approach, to study the statistical sensitivities of different combinations of future surveys, we
perform a prospective analysis based on the Fisher matrix technique (Fisher 1935). We validate this approach by comparing its
estimates of the statistical errors for the current data set with those obtained with the frequentist method described above.
The statistical errors on the n cosmological parameters θ = (θ1 , . . . , θn ) are determined by using the inverse of the covariance
matrix V called the Fisher matrix F defined as:
(V −1 )i j = Fi j = −
∂2 ln L(x; θ)
,
∂θi ∂θ j
(9)
where L(x; θ) is the likelihood function depending on the n cosmological parameters and a data set of N measured quantities
x = (x1 , . . . , xN ). A lower bound, and often a good estimate, for the statistical error on the cosmological parameter θ i is given by
(Vii )1/2 .
When the measurements of several cosmological probes are combined, the total Fisher matrix F tot is the sum of the three
Fisher matrices F S N , FWL and FCMB corresponding respectively to the SNIa, weak lensing and CMB observations. The total
−1
covariance matrix F tot
allows us to estimate both, the expected sensitivity on the cosmological parameters, with the diagonal
terms, and the correlations between the parameters, with the off-diagonal terms. The Fisher matrices for each probe are computed
as follows.
CMB: In the case of CMB experiments, the data set vector x corresponds to the measurements of C ` , the angular power spectrum
of the CMB from ` = 2 to some cutoff `max . Using Eq. 9, the Fisher matrix is written as
(FCMB )i j =
`max
X
1 ∂C` ∂C`
·
·
2
∂θi ∂θ j
σ
C`
l=2
where σC` is the statistical error on C ` obtained directly from published results or estimated as (see Knox 1995):
s
#
"
`2 θ2
f whm
2
2
8
ln(2)
σC` =
C` + (θ f whm s) · e
(2` + 1) f sky
(10)
(11)
where the second term incorporates the effects of instrumental noise and beam smearing. In Eq. 11, θ f whm , f sky , and s are respectively the angular resolution, the fraction of the sky observed and the expected sensitivity per pixel.
The C` and their derivatives with respect to the various cosmological parameters are computed with CMBEASY (Doran
2003), an object oriented C++ package derived from CMBFAST (Seljak & Zaldarriaga 1996).
SNIa: The SNIa apparent magnitudes m can be expressed as a function of the luminosity distance as
m(z) = M s0 + 5log10 (DL )
(12)
Ch. Yèche et al.: Prospects for Dark Energy Evolution
5
where DL (z) ≡ (H0 /c) dL (z) is the H0 -independent luminosity distance to an object at redshift z. The usual luminosity distance
dL (z) is related to the comoving angular-diameter distance r(χ) by d L (z) = (1 + z) · r(χ), with the definition of r(χ) and χ(z) given
in Sec. 2.1. The normalisation parameter M s0 thus depends on H0 and on the absolute magnitude of SNIa.
The Fisher matrix, in this case, is related to the measured apparent magnitude m k of each object and its statistical error σmk
by
X 1 ∂mk ∂mk
(FS N )i j =
·
.
(13)
·
σ2mk ∂θi ∂θ j
k
Weak lensing: The weak lensing power spectrum is given by (e.g., Hu & Tegmark 1999, cf, Refregier 2003 for conventions)
"
#2
!
Z
9 H 0 4 2 χh
g(χ)
`
C` =
Ωm
dχ
P ,χ ,
(14)
16 c
ar(χ)
r
0
where r(χ) is the comoving angular-diameter distance, and χh corresponds to the comoving distance to horizon. The radial weight
function g is given by
Z χh
r(χ)r(χ0 − χ)
dχ0 n(χ0 )
,
(15)
g(χ) = 2
r(χ0 )
χ
R
where n(χ) is the probability of finding a galaxy at comoving distance χ and is normalised as dχ n(χ) = 1.
The linear matter power spectrum P(k, z) is computed using the transfer function from Bardeen et al. (1986) with the conventions of Peacock (1997), thus ignoring the corrections on large scales for quintessence models (Ma et al. 1999). The linear
growth factor of the matter overdensities δ is given by the well known equation:
3
δ̈ + 2H δ̇ − H 2 Ωm (a)δ = 0,
2
(16)
where dots correspond to time derivatives, and Ωm (a) is the matter density parameter at the epoch corresponding to the dimensionless scale factor a. This equation is integrated numerically with boundary conditions given by the matter-dominated solution,
G = δ/a = 1 and Ġ = 0, as a → 0 (see eg. Linder & Jenkins 2003). We enforce the CMB normalisation of the power spectrum
P(k, 0) at z = 0 using the relationship between the WMAP normalisation parameter A and σ 8 given by Hu (2004). Considerable
uncertainties remain for the non-linear corrections in quintessence models (cf. discussion in Hu (2002)). Here, we use the fitting
formula from Peacock & Dodds (1996).
For a measurement of the power spectrum, the Fisher matrix element is defined as:
X 1 ∂C` ∂C`
(FWL )i j =
,
(17)
σC2 ∂θi ∂θ j
`
`
where the summation is over modes ` which can be reliably measured. This expression assumes that the errors σ C` on the lensing
power spectrum are gaussian and that the different modes are uncorrelated. Mode-to-mode correlations have been shown to
increase the errors on cosmological parameters (Cooray & Hu 2001) but are neglected in this paper.
Neglecting non-gaussian corrections, the statistical error σC` in measuring the lensing power spectrum C ` (cf., e.g.,
Kaiser 1998, Hu & Tegmark 1999, Hu 2002) is given by:
s



σ2γ 
2
σC` =
(18)
Cl +
 ,
(2l + 1) fsky
2ng 
where fsky is the fraction of the sky covered by the survey, ng is the surface density of usable galaxies, and σ2γ = h|γ|2 i is the shear
variance per galaxy arising from intrinsic shapes and measurement errors.
2.3. Cosmological parameters and models
For the studies presented in this paper, we limit ourselves to the 9 cosmological parameters: θ = Ω b , Ωm , h, n s, τ, w0 , wa , A
and M s0 , with the following standard definitions:
- (Ωi , i=b,m) are densities for baryon and matter respectively (Ωm includes both dark matter and baryons),
- h is the Hubble constant in units of 100 km/s/Mpc,
- n s is the spectral index of the primordial power spectrum,
- τ is the reionisation optical depth,
- A is the normalisation parameter of the power spectrum for CMB and weak lensing (cf Hu & Tegmark (1999) for definitions).
The matter power spectrum is normalised according to the COBE normalisation (Bunn & White 1997), which corresponds
6
Ch. Yèche et al.: Prospects for Dark Energy Evolution
to σ8 = 0.88. This is consistent with the WMAP results (Spergel et al. 2003) and with the average of recent cosmic shear
measurements (see compilation tables in Mellier et al. 2002, Hoekstra et al. 2002, Refregier 2003).
- M s0 is the normalisation parameter from SNIa (cf Sec. 2.2.2),
- Dark energy is described by the w0 parameter corresponding to the value of the equation of state at z=0. When the z dependence
of the equation of state is studied, an additional parameter wa is defined (cf Sec. 2.1).
The reference fiducial model of our simulation is a ΛCDM model with parameters Ωm = 0.27, Ωb = 0.0463, n s = 0.99,
h = 0.72, τ = 0.066, A = 0.86, consistent with the WMAP experiment (see tables 1-2 in Spergel et al. 2003). In agreement with
this experiment, we assume throughout this paper that the universe is flat, i.e., Ω = Ω m + ΩX = 1. We also neglect the effect of
neutrinos, using 3 degenerate families of neutrinos with masses fixed to 0.
In the following, we will consider deviations from this reference model. For the equation of state, we use as a reference
w0 = −0.95 and wa = 0 as central values (we have not used exactly w0 = −1 to avoid transition problems in the CMB
calculations). To estimate the sensitivity on the parameters describing the equation of state, we also consider two other fiducial
models: a SUGRA model, with (w0 = −0.8, wa = 0.3) as proposed by, e.g., Weller & Albrecht (2002) to represent quintessence
models, and a phantom model (Caldwell 2002) with (w0 = −1.2, wa = −0.3).
In this analysis, the full covariance matrix on all parameters is used with no prior constraints on the parameters, avoiding
biases from internal degeneracies. We have implemented the time evolving parametrization of the equation of state in simulations
and analysis of the three probes we consider in this paper, i.e. CMB, SNIa and weak lensing.
3. Combination of current surveys
We first apply our statistical approach to the combination of recent SNIa and CMB data, without any external constraints
or priors. The comparison of the statistical errors obtained with a global fit using this frequentist treatment, with those predicted
with the Fisher matrix technique, also allows us to validate the procedure described in Sec. 2. Finally, we compare our results
with other published results.
3.1. Current surveys
We use the ’Gold sample’ data compiled by Riess et al. (2004), with 157 SNIa including a few at z > 1.3 from the Hubble
Space Telescope (HST GOODS ACS Treasury survey), and the published data from WMAP taken from Spergel et al. (2003).
We perform two distinct analyses: in the first case, the equation of state is held constant with a single parameter w0 and we fit
8 parameters, as described in Sec. 2.2; in the second case, the z dependence of the equation of state is modelled by two variables
w0 and wa as defined in Sec. 2.1, and we fit 9 parameters.
3.2. Results
The results of this frequentist combination of CMB and SNIa data are summarised in Tab. 1. When the equation of state
+0.10
is considered constant, we obtain w0 = −0.92−0.13
(1-σ) and the shape of the CL is relatively symmetrical around the value
of w0 obtained at the χ2 minimum. When a z dependence is added to the equation of state, the CL is still symmetrical with
+0.13
w0 = −1.09−0.15
but wa becomes asymmetrical with a long tail for smaller values of wa , as can be seen in Fig. 1. The 1-D CL for
+0.21 +0.42
wa gives the resulting CL at 68%(1σ) and 95%(2σ): wa = 0.82−0.26
−0.80 .
Tab. 1 compares the 1σ errors obtained with the frequentist method and the errors predicted with the Fisher matrix techniques.
The agreement is good, and in the remaining part of this paper, for the combination of expectations from future surveys, we will
use the Fisher matrix approach.
However Upadhye et al. (2004) noticed that the high redshift limit of the parametrization of the EOS plays an important role
when we consider CMB data which impose w(z → ∞) < 0. With our choice of parametrization (see definition in Eq. 3), we get
the condition w0 + wa < 0. When a fit solution is found close to this boundary condition, as is the case with the current data, the
CL distributions are asymmetric, giving asymmetrical errors. The Fisher matrix method is not able to represent complicated 2-D
CL shapes, as those shown in Fig. 2. For example, the error on w a increases when the (w0 , wa ) solution moves away from the
’unphysical’ region w0 + wa > 0. To avoid this limitation, we will thus use fiducial values of wa closer to zero for the prospective
studies with future surveys.
It is worth noting that the solution found by the fit corresponds to a value of w slightly smaller than -1 for z = 0, and a value
of w slightly larger than -1 for high z. The errors are such that the value of w is compatible with -1. However, this technically
means that the Universe crosses the phantom line in its evolution. This region (w < −1) cannot be reached by the fit, if dark
energy perturbations are computed in the CMBEASY version we use. To obtain a solution and compare with other published
results, we therefore probed two different conditions, both illustrated in Fig. 2.
Ch. Yèche et al.: Prospects for Dark Energy Evolution
7
Table 1. Results of the frequentist fit to WMAP and Riess et al. (2004) SNIa data. For the 8 parameter fit with a constant EOS, the first column
gives the value of the variable at the χ2 minimum, with the confidence interval at 68% (1 σ), the second column shows the 1σ error computed
with the Fisher matrix techniques. The third and fourth columns present the same information for the 9 parameter fit with a z dependent EOS.
+0.42
The 1σ errors are symmetrical for all the variables except for wa . Its error goes from +0.21
−0.26 for CL at 68% to −0.80 for CL at 95% (see text).
1
1-CL
1-CL
Ωb
Ωm
h
nS
τ
w0
wa
A
Ms0
constant EOS
fit
σFisher
0.049+0.005
±0.003
−0.003
0.29+0.05
±0.04
−0.04
0.69+0.03
±0.03
−0.02
0.97+0.03
±0.03
−0.03
0.13+0.04
±0.04
−0.04
−0.92+0.10
±0.11
−0.13
0.79+0.08
±0.10
−0.07
15.94+0.03
±0.03
−0.03
0.9
z dependent EOS
fit
σFisher
0.055+0.003
±0.003
−0.003
0.33+0.04
±0.04
−0.04
0.69+0.03
±0.03
−0.02
0.97+0.03
±0.03
−0.03
0.14+0.04
±0.04
−0.04
−1.09+0.13
±0.14
−0.15
0.82+0.21
±0.25
−0.26
0.80+0.08
±0.10
−0.07
15.95+0.03
±0.03
−0.03
1
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
-1.4
-1.3
-1.2
-1.1
-1
-0.9
-0.8
w0
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
wa
Fig. 1. Confidence level (CL) plots on parameters w0 (left) and wa (right), using WMAP and Riess et al. 2004 SNIa data for a 9 parameter fit
with evolving EOS. The dashed lines correspond to the 68%(1σ) and 90%(1.64σ) confidence intervals.
First, we removed altogether the perturbations for the dark energy, which gives the results presented above. This allows a
comparison with Seljak et al. (2004), who have likely removed dark energy perturbations. Their central value corresponds to
+1.92
+0.38
at 95%(2σ). It is closer to w = −1 than our result and gives errors for wa larger than the
w0 = −0.98−0.37
and wa = −0.05−1.13
ones we get. The comparison is however not exact, since Seljak et al. use a bayesian approach for the fits, and give results for
an evolving equation of state, only for the total combination of the WMAP and SNIa data with other SDSS probes (galaxies
clustering, bias, and Lyman α forest).
We also performed the fits, including dark energy perturbations, only when w > −1 (which is the default implementation in
CMBFAST). Caldwell & Doran (2005) have argued convincingly that crossing the cosmological constant boundary leaves no
distinct imprint, i.e., the contributions of w < −1 are negligible, because w < −1 dominates only at late times and dark energy
does not generally give strong gravitational clustering. Our analysis, including dark energy perturbations only when w > −1,
+0.15
+0.5
gives a minimum (cf. right hand side plot in Fig. 2) for w0 = −1.32−0.19
and wa = 1.2−0.8
at 1σ. This is some 2σ away from
the no perturbation case. We remark that these values are very close to those obtained by Upadhye et al.(2004), who use a
procedure similar to ours, without any marginalisation on parameters, a weak constraint w 0 + wa ≤ 0 inside their fit. Their result,
+0.40
+0.34
at 95%(2σ), has almost the same central value as our fit, when we switch on the dark energy
and wa = 1.25−2.17
w0 = −1.3−0.39
perturbation for w > −1. The errors we get are also compatible, and are much larger than in the no perturbation case.
The importance and impact of introducing dark energy perturbations has been discussed by Weller & Lewis (2003). Their
combined WMAP and SNIa analysis with a constant sound speed also gives a more negative value of w, when a redshift
dependence is taken into account. Although Rapetti et al. (2004) observe a reduced effect when they add cluster data, they still
indicate a similar trend. Finally, when dark energy perturbations are included, we observe that the minimisation is more difficult
and correlations between parameters increase.
We conclude that our results are compatible with other published papers using various combinations of cosmological probes.
There is a good agreement of all analysis when w0 is constant, showing that data agree well with the ΛCDM model. However,
2.5
0.9
wa
Ch. Yèche et al.: Prospects for Dark Energy Evolution
wa
8
2.5
0.9
2
0.8
2
0.8
1.5
0.7
1.5
0.7
0.6
1
0.6
1
0.5
0.5
0.4
0
-0.5
0.3
No DE
0.2
Perturbation
0.4
0
-0.5
0.1
-1
-2.2
0.5
0.5
0.3
With DE
0.2
Perturbations
0.1
-1
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
w0
0
-2.2
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
0
w0
Fig. 2. Confidence level contour plots with WMAP and Riess et al. 2004 SNIa data, for the 9 parameter fit with a z dependent EOS in the plane
(w0 , wa ). The plot on the left hand side corresponds to the case when we introduce no dark energy perturbation. For the plot on the right hand
side, we introduce dark energy perturbations only when w > −1.
large uncertainties remain for the location of the minimum in the (w0 , wa ) plane, when a redshift variation is allowed. We
emphasise that this is not due to the statistical method but to internal assumptions. Upadhye et al.(2004) mention the sensitivity
to the choice of parametrization. We show that the introduction of dark energy perturbations for w > −1, can change the minimum
by nearly 2σ and that the minimum is not well established as correlations between parameters increase, and errors, in this zone
of parameter space are very large.
For the sake of simplicity, we decided to present, in the rest of this paper, a prospective study without dark energy perturbations, using a Fisher matrix technique.
4. Combination of future surveys
In this section, we study the sensitivity of the combination of future CMB, SNIa and weak lensing surveys for dark energy
evolution. We expect new measurements from the CHTLS surveys in SNIa and weak lensing in the next few years, which can
be combined with the first-year WMAP together with the expected CMB data from the Olimpo CMB balloon experiment. These
are what we call ’mid term’ surveys.
The combined mid term results will be compared to the ’long term’ expectations from the next generation of observations in
space which are under preparation, i.e., the Planck Surveyor mission for CMB, expected in 2007, and the SNAP/JDEM mission,
a large imaging survey, expected for 2014, which includes both SNIa and weak lensing surveys.
4.1. Mid term surveys
The different assumptions we use for the mid term simulations are as follows, and are summarised in Tab. 3.
CMB: We add to the WMAP data, some simulated CMB expectations from the Olimpo balloon experiment (Masi et al. 2003),
equipped with a 2.6 m telescope and 4 bolometers arrays for frequency bands centered at 143, 220, 410 and 540 GHz. This
experiment will also allow us to observe the first ”large” survey of galaxies cluster through the SZ effect. For this paper, we will
limit our study to CMB anisotropy aspects.
For a nominal 10 days flight with an angular resolution θ f whm = 40 and with f sky ' 1%, the expected sensitivity per pixel is
s = 3.4 × 10−6 . We use Eq. 11 to estimate the statistical error σC` on the angular power spectrum.
SNIa: We simulate future SNIa measurements derived from the large SNLS (2001) ground based survey within the CFHTLS
(2001). This survey has started in 2003 and expects to collect a sample of 700 identified SNIa in the redshift range 0.3 < z < 1, after 5 years of observations. We simulate the sample, as explained in Virey et al. (2004a) with the number of SNIa shown in Tab. 2,
in agreement with the expected SNIa rates from SNLS. We assume a magnitude dispersion of 0.15 for each supernova, constant
Ch. Yèche et al.: Prospects for Dark Energy Evolution
9
in redshift after all corrections. This uncertainty corresponds to the most favourable case in which experimental systematic errors
are not considered.
A set of 200 very well calibrated SNIa at redshift < 0.1 should be measured by the SN factory (Wood-Vasey et al. 2004)
project. This sample is needed to normalise the Hubble diagram and will be called the ’nearby’ sample.
Table 2. Number of simulated SNIa by bins of 0.1 in redshift for SNLS+HST and SNAP respectively.
z
SNLS + HST
SNAP
0.2
35
0.3
44
64
0.4
56
95
0.5
80
124
0.6
96
150
0.7
100
171
0.8
104
183
0.9
108
179
1.0
10
170
1.1
14
155
1.2
7
142
1.3
12
130
1.4
5
119
1.5
2
107
1.6
3
94
1.7
1
80
Finally, to be as complete as possible, we simulate a set of 54 SNIa, expected from HST programs, with a magnitude dispersion of 0.17 for each supernova, at redshifts between 1 and 1.7. Tab. 3 summarises the simulation parameters.
Weak lensing: The coherent distortions that lensing induces on the shape of background galaxies have now been firmly measured
from the ground and from space. The amplitude and angular dependence of this ‘cosmic shear’ signal can be used to set strong
constraints on cosmological parameters.
Earlier studies of the constraints on dark energy from generic weak lensing surveys can be found in Hu & Tegmark (1999),
Huterer (2001), Hu (2002). More recently, predictions for the constraints on an evolving w(a) were studied by several authors
(e.g., Benabed & van Waerbeke 2004, Lewis & Bridle 2002). We expect, in the near future, new cosmic shear results from the
CFHTLS wide survey (CFHTLS 2001).
In this paper, we will consider measurements of the lensing power spectrum C ` with galaxies in two redshift bins. We will
only consider modes between ` = 10 and 20000, thus avoiding small scales where instrumental systematics and theoretical
uncertainties are more important.
For the CFHTLS survey, we assume a sky coverage of 170◦2 . The rms shear error per galaxy is taken as σγ = 0.35 and the
surface density of usable galaxies as 20 amin−2 which is divided evenly into to redshift bins with median redshifts z m = 0.72 and
1.08. The redshift distribution of the galaxies in each redshift bin is taken to be as in Bacon et al. (2000) with the above median
redshifts (cf Tab. 3 for a summary of the survey parameters). We use Eq. 18 to estimate the statistical error σ C` .
4.2. Long term survey
The future will see larger surveys both from the ground and space. To estimate the gain for large ground surveys compared
to space, critical studies taking into account the intrinsic ground limitation (both in distance and in systematics) should be done,
and systematic effects, not included here, will be the dominant limitation. In this paper, we limit ourselves to the future space
missions.
We simulate the Planck Surveyor mission using Eq. 11 with the performances described in Tauber et al. (2004). Assuming
that the other frequency bands will be used to identify the astrophysical foregrounds, for the CMB study over the whole sky, we
consider only the three frequency bands (100, 143 and 217 GHz) with respectively (θ f whm = 9.20 , 7.10 and 5.00 ) resolution and
(s = 2.0 10−6, 2.2 10−6 and 4.8 10−6) sensitivity per pixel.
We also simulate observations from the future SNAP satellite, a 2 m telescope which plans to discover around 2000 identified
SNIa, at redshift 0.2< z <1.7 with very precise photometry and spectroscopy. The SNIa distribution, given in Tab. 2, is taken
from Kim et al. (2004). The magnitude dispersion σ(m)disp is assumed to be 0.15, constant and independent of the redshift, for
all SNIa after correction. Moreover, we introduce an irreducible systematic error σ(m) irr following the prescription of Kim et al.
(2004). In consequence, the total error on the magnitude σ(m) tot per redshift bin i, is defined as: σ(m)2tot,i = σ(m)2disp /Ni + σ(m)2irr
where Ni is the number of SNIa in the ith 0.1 redshift bin. In the case of SNAP, σ(m) irr is equal to 0.02.
The SNAP mission also plans a large cosmic shear survey. The possibilities for the measurement of a constant equation of
state parameter w with lensing data were studied by Rhodes et al. (2004), Massey et al. (2004), Refregier et al. (2004). We extend
here the study in the case of an evolving equation of state. We use in the simulation the same assumptions as in Refregier et al.
(2004) with a measurement of the lensing power spectrum in 2 redshift bins, except for the survey size, which has increased from
300◦2 to 1000◦2 (Aldering et al. 2004) and for the more conservative range of multipoles ` considered (see §4.1).
The long term survey parameters are summarised in Tab. 3.
4.3. Results
The combination of the three data sets is performed with, and without, a redshift variation for the equation of state, for both
mid term and long term data sets.
10
Ch. Yèche et al.: Prospects for Dark Energy Evolution
Table 3. Simulation inputs for CMB, SNIa and Weak Lensing observations
Current
Data
WMAP (Spergel et al.(2003))
Mid term
Data
Olimpo
+ WMAP
Long term
data
Planck
Current
Data
Riess et al. (2004) + HST
Mid term
Data
SNfactory
SNLS
HST
Long term
Data
SNfactory
SNAP
Mid term
Data
CFHTLS
Long term
Data
SNAP
CMB surveys
fsky
f(GHz)
full sky
23/33/41/61/94
θfwhm (0)
13
s(10−6 )
-
0.01
143/220/410/540
4
3.4
full Sky
100
143
217
9.2
7.1
5.0
2.0
2.2
4.8
Statistical error
∼ 0.25
Systematic error
-
0.15
0.15
0.17
-
SN surveys
SN #
Redshift range
157
z < 1.7
200
700
54
z < 0.1
0.3 < z < 1
1<z
300
z < 0.1
2000
0.1 < z < 1.7
WL surveys
zm (2 bins)
A(deg2 )
0.72, 1.08
170
0.95, 1.74
1000
0.15
0.15
0.02
total ng (amin−2 )
20
σγ
0.35
100
0.31
The different plots in Fig. 3 show the results for individual mid term probes and for their combination. The results are for a
constant w0 , plotted as a function of the matter density Ωm . The combined contours are drawn using the full correlation matrix
on the 8 parameters for the different sets of data.
The SNLS survey combined with the nearby sample will improve the present precision on w by a factor 2. The expected
contours from cosmic shear have the same behaviour as the CMB but provide a slightly better constraint on Ω m and a different
correlation with w: CMB and weak lensing data have a positive (w, Ω m ) correlation compared to SNIa data, which have a negative
correlation. This explains the impressive gain when the three data sets are combined, as shown in Tab. 4. Combining WMAP
with Olimpo data, helps to constrain w through the correlation matrix as Olimpo expects to have more information for the large
` of the power spectrum.
Fig. 4 gives the expected accuracy of the mid term surveys on the parameters of an evolving equation of state. The CL
contours plots of wa versus w0 , are obtained with a 9 cosmological parameter fit. Here also, we observe a good complementarity:
there is little information on the time evolution from SNIa with no prior, while the large redshift range from CMB data is adding
a strong anti-correlated constraint on wa .
A combined analysis proves far superior to analysis with only SNIa. In the favourable case, where we add more SNIa from
HST survey, we expect a gain of a factor 2 on the errors, but it is not enough to lift degeneracies and the expected precision on
wa with these data will not be sufficient to answer questions on the nature of the dark energy.
The simulated future space missions show an improved sensitivity to the time evolution of the equation of state. The accuracy
on wa for the different combinations are summarised in Tab. 4. There is again a large improvement from the combination of the
three data sets. The precision, for the long term surveys, will be sufficient to discriminate between the different models we have
chosen, as shown in the left hand side plot of Fig. 5 and in Tab. 5, while it is not the case for the mid term surveys. This figure
illustrates, moreover, that the errors on wa and w0 , and the correlation between these two variables are strongly dependent on the
choice of the fiducial model.
More generally, the combination of the probes with the full correlation matrix allows the extraction of the entire information
available. For instance, the large correlation between nS and wa observed for the weak lensing probe combined with the precise
measurement of n s given by the CMB, gives a better sensitivity on wa than the simple combination of the two wa values, obtained
separately for the CMB and weak lensing. Such an effect occurs for several other pairs of cosmological parameters considered
Ch. Yèche et al.: Prospects for Dark Energy Evolution
11
Fig. 3. CL contours for mid term CMB (WMAP +Olimpo), SNIa and weak lensing data from CFHTLS and the combination of the three probes
for the 8 parameter fit in the plane (Ωm , w0 ) (see also Tab. 4). The solid lines represent 68% (1 σ), 95% (2σ), and 99% CL contours.
Table 4. Expected sensitivity on cosmological parameters for three scenarii: Current supernova and CMB experiments (WMAP and Riess et
al.2004), mid term experiments (CFHT-SNLS (supernova surveys), CFHTLS-WL (weak lensing) and CMB (WMAP+Olimpo)), long term
experiments (CMB (Planck) and SNAP (supernovae and weak lensing)). For each scenario, the first column gives the 1σ error computed
with the Fisher matrix techniques for the 8 free parameter configuration and the second columns gives the 1σ error for the 9 free parameter
configuration.
Scenario
Ωb
Ωm
h
ns
τ
w0
wa
A
Ms0
Today
0.003 0.004
0.04
0.04
0.03
0.03
0.03
0.03
0.05
0.04
0.11
0.22
−
0.99
0.10
0.10
0.03
0.03
Mid term
0.001 0.002
0.01
0.01
0.01
0.01
0.006 0.009
0.01
0.01
0.02
0.10
−
0.43
0.02
0.02
0.01
0.01
Long Term
0.0008 0.0008
0.004
0.004
0.006
0.006
0.003
0.003
0.01
0.01
0.02
0.04
−
0.07
0.02
0.02
0.01
0.01
in this study. The plot, in the right hand side of Fig 5, is an illustration of this effect. It shows the combination of the 3 probes in
the (w0 , wa ) plane. The 1σ contour for the combined three probes, is more constraining than the 2-D combination in the (w 0 , wa )
plane of the three probes.
Finally, in the long term scenario, the weak lensing probe provides a sensitivity on the measurement of (w 0 , wa ) comparable
with those of the combined SN and CMB probes, whereas in the mid term scenario the information brought by weak lensing
was marginal. This large improvement observed in the information provided by the weak lensing, can be explained by the larger
survey size and the deeper volume probed by SNAP/JDEM, compared to the ground CFHTLS WL survey. We thus conclude
that adding weak lensing information will be an efficient way to help distinguishing between dark energy models. If systematic
effects are well controlled, the future dedicated space missions may achieve a sensitivity of order 0.1 on w a .
12
Ch. Yèche et al.: Prospects for Dark Energy Evolution
Fig. 4. CL contours for mid term CMB (WMAP +Olimpo), SNIa and weak lensing data from CFHTLS and the combination of the three probes
for the 9 parameter fit in the plane (w0 , wa ) (see also Tab. 4). The solid lines represent 68% (1 σ), 95% (2σ), and 99% CL contours.
Table 5. Expected sensitivity on cosmological parameters for the long term missions with CMB (Planck) and SNAP (supernova surveys and
weak lensing) for the 9 free parameter configuration.
Model
Ωb
Ωm
h
nS
τ
w0
wa
A
Ms0
ΛCDM
0.0008
0.004
0.006
0.003
0.01
0.04
0.07
0.02
0.015
SUGRA
0.0008
0.004
0.006
0.003
0.01
0.04
0.06
0.02
0.014
Phantom
0.0007
0.003
0.005
0.003
0.01
0.03
0.14
0.02
0.013
The SNAP/JDEM space mission is designed, in principle, to control its observational systematic effects for SNIa to the %
level, which is probably impossible to reach for future ground experiments. In this study, we assign an irreducible systematic
error on SNIa magnitudes of 0.02 and systematic effects have been neglected for CMB and weak lensing. This can have serious
impacts on the final sensitivity, in particular, on the relative importance of each probe.
Other probes, whose combined effects we have not presented in this paper, but intend to do in forthcoming studies, remain
therefore most useful. For example, the recent evidence for baryonic oscillations (Eisenstein et al. 2005) is a proof that new
probes can be found. The present constraints that these results provide, do not improve the combined analysis we present here.
However, getting similar results from different probes greatly contributes to the credibility of a result, in particular, when the
systematical effects can be quite different, as is the case for the different probes we consider. Finally, the joint analysis of cluster
data observed simultaneously with WL, SZ effect and X-rays, will allow the reduction of the intrinsic systematics of the WL
probe.
1
0.6
wa
wa
Ch. Yèche et al.: Prospects for Dark Energy Evolution
13
1
0.6
0.9
0.4
0.9
0.4
0.8
0.2
0.7
0.6
-0
0.8
0.2
0.7
0.6
-0
0.5
-0.2
0.4
0.3
-0.4
0.5
-0.2
-0.4
0.2
-0.6
0.3
WL
0.2
SN + CMB
0.1
-0.6
0.1
-0.8
-1.3
0.4
SN + CMB + WL
-1.2
-1.1
-1
-0.9
-0.8
-0.7
-0.6
0
-0.8
-1.3
-1.2
-1.1
-1
w0
-0.9
-0.8
-0.7
-0.6
0
w0
Fig. 5. CL contours for future space data from SNAP (SNIa and WL) and Planck (CMB) for a 9 parameter fit in the plane (w 0 , wa ). The left hand
side figure shows the combination of SNAP (SNIa+WL) and CMB for three different models (ΛCDM, SUGRA and Phantom). The solid lines
represent 68% (1 σ), 95% (2σ), and 99% CL contours. The right hand side figure shows the CL for the combined three ”long term” probes.
The solid lines are the 1σ contours for different combinations: WL alone, combined SNIa and CMB, and the three combined probes.
5. Conclusions
In this paper, we have presented a statistical method based on a frequentist approach to combine different cosmological
probes. We have taken into account the full correlations of parameters without any priors, and without the use of Markov chains.
Using current SNIa and WMAP data, we fit a parametrization of an evolving equation of state and find results in good
agreement with other studies in the literature. We confirm that data prefer a value of w less than -1 but are still in good agreement
with the ΛCDM model. We emphasise the impact of the implementation of the dark energy perturbations. This can explain the
discrepancies in the central values found by various authors. We have performed a complete statistical treatment, evaluated the
errors for existing data and validated that the Fisher matrix technique is a reliable approach as long as the parameters (w 0 , wa ) are
in the ‘physical’ region imposed by CMB boundary condition: w(z → ∞) < 0.
We have then used the Fisher approximation to calculate the expected errors for current surveys on the ground (e.g., CFHTLS)
combined with CMB data, and compared them with the expected improvements from future space experiments. We confirm that
the complete combination of the three probes, including weak lensing data, is very powerful for the extraction of a constant w.
However, a second generation of experiments like the Planck and SNAP/JDEM space missions is required, to access the variation
of the equation of state with redshift, at the 0.1 precision level. This level of precision needs to be confirmed by further studies
of systematical effects, especially for weak lensing.
Acknowledgements. The authors are most grateful to M. Doran for the CMBEASY package, the only code that was not developed by this
collaboration, and for his readiness to answer all questions. They wish to thank A. Amara, J. Berg´e, A. Bonissent, D. Fouchez, F. HenryCouannier, S. Basa, J.-M. Deharveng, J.-P. Kneib, R. Malina, C. Marinoni, A. Mazure, J. Rich, and P. Taxil for their contributions to stimulating
discussions.
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2.3.2 Contraintes sur la courbure et une energie noire dynamique
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Article publie sous la reference : Phys. Lett. B648 (2007) 8 .
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1.5
w = w0 + w1(1−a)
w = −1
w = w0 + w1sin(w2ln(a))
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2.4
Test cosmologique a partir de la cinematique des
galaxies
Dans cette partie nous proposons un nouveau test cosmologique a partir des proprietes
cinematiques des galaxies. L'idee originale, essentiellement due a Christian Marinoni, est
d'utiliser les vitesses particulieres des galaxies an de construire des \regles" standard ainsi
que des chandelles standard.
Il est connu depuis longtemps (Tully & Fisher 1977 [99, 100]) qu'il existe une relation,
basee sur les observations, entre la vitesse de rotation d'une galaxie spirale et son diametre,
ainsi qu'avec sa luminosite. Du point de vue theorique, on s'attend a l'existence d'une telle
relation dans le cadre des modeles de formation des structures avec matiere noire froide
[101]. La relation de Tully-Fisher est utilisee localement pour determiner les distances des
galaxies et la constante de Hubble H0.
Dans les articles qui suivent il est propose d'utiliser cette relation dans un contexte
cosmologique an de denir des \regles" standard et des \chandelles" standard a haut
redshift. Plus precisemment, la relation de Tully-Fisher nous indique que des galaxies
spirales possedant la m^eme vitesse de rotation ont statistiquement la m^eme taille et la
m^eme luminosite. L'expansion de l'univers a un eet direct sur l'evolution dans le temps
(i.e. en redshift) de ces proprietes physiques. On peut alors utiliser les catalogues de
galaxies a haut redshift, tel que ceux fournis par les collaborations VVDS [102](disponible)
ou zCOSMOS [103] (en cours de mesure), pour tenter de mettre en application ce test
cosmologique et contraindre les parametres du fond (
M , X et wX ).
Malheureusement, la situation se complique lorsque l'on prend en compte les evolutions
dans le temps des diametres et des luminosites dues a la physique interne des galaxies. On
obtient alors une correspondance entre les eets cosmologiques et les eets d'evolution intrinseque. On s'est aussi interesse aux biais sur les parametres cosmologiques que peuvent
entrainer les eets d'evolution intrinseque et aux moyens de reconnaitre sans ambiguites
ses eets.
Le premier article, relativement theorique, presente l'idee de base. Le second applique
cette idee aux mesures realisees par la collaboration VVDS qui a realise un sondage profond
de galaxies avec une spectroscopie tres precise. Ces deux articles seront publies prochainement dans Astronomy & Astrophysics.
217
Astronomy & Astrophysics manuscript no. aa7116-07
September 7, 2007
c ESO 2007
Geometrical tests of cosmological models
I. Probing dark Energy using the kinematics of high redshift galaxies
C. Marinoni1 , A. Saintonge2 , R. Giovanelli2 , M. P. Haynes2 , K. L. Masters3 , O. Le Fèvre4 , A. Mazure4 ,
P. Taxil1 , and J.-M. Virey1
1
2
3
4
Centre de Physique Théorique , CNRS-Université de Provence, Case 907, 13288 Marseille, France
e-mail: marinoni@cpt.univ-mrs.fr
Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02143, USA
Laboratoire d’Astrophysique de Marseille, UMR 6110, CNRS Université de Provence, 13376 Marseille, France
Received 17 January 2007 / Accepted 2 August 2007
ABSTRACT
We suggest to use the observationally measured and theoretically justified correlation between size and rotational velocity of galactic
discs as a viable method to select a set of high redshift standard rods which may be used to explore the dark energy content of the
universe via the classical angular-diameter test. Here we explore a new strategy for an optimal implementation of this test. We propose
to use the rotation speed of high redshift galaxies as a standard size indicator and show how high resolution multi-object spectroscopy
and ACS/HST high quality spatial images, may be combined to measure the amplitude of the dark energy density parameter ΩQ ,
or to constrain the cosmic equation of state parameter for a smooth dark energy component (w = p/ρ, −1 ≤ w < −1/3). Nearly
1300 standard rods with high velocity rotation in the bin V = 200 ± 20 km s−1 are expected in a field of 1 sq. degree and over the
redshift baseline 0 < z < 1.4. This sample is sufficient to constrain the cosmic equation of state parameter w at a level of 20% (without
priors in the [Ωm , ΩQ ] plane) even when the [OII]λ3727 Å linewidth-diameter relationship is calibrated with a scatter of ∼40%.
We evaluate how systematics may affect the proposed tests, and find that a linear standard rod evolution, causing galaxy dimensions
to be up to 30% smaller at z = 1.5, can be uniquely diagnosed, and will minimally bias the confidence level contours in the [ΩQ ,
w] plane. Finally, we show how to derive, without a priori knowing the specific functional form of disc evolution, a cosmologyevolution diagram with which it is possible to establish a mapping between different cosmological models and the amount of galaxy
disc/luminosity evolution expected at a given redshift.
Key words. cosmological parameters – cosmology: observations – cosmology: theory – cosmology: cosmological parameters –
galaxies: high-redshift – galaxies: fundamental parameters – galaxies: evolution
1. Introduction
Several and remarkable progresses in the understanding of the
dynamical status of the universe, encourage us to believe that,
after roaming from paradigm to paradigm, we are finally converging towards a well-founded, internally consistent standard
model of the universe.
The picture emerging from independent observations and
analysis is sufficiently coherent to be referred to as the concordance model (e.g. Tegmark 2006). Within this framework, the
universe is flat (ΩK = −0.003+0.0095
−0.0102 ) composed of ∼1/5 cold
dark matter (Ωcdm ∼ 0.197+0.016
−0.015 ) and ∼3/4 dark energy (ΩΛ =
0.761+0.017
),
with
large
negative
pressure (w = −0.941+0.017
−0.018
−0.018),
and with a very low baryon content (Ωb = 0.0416+0.0019
−0.0018).
Mounting and compelling evidence for accelerated expansion
of the universe, driven by a dark energy component, presently
relies on our comprehension of the mechanisms with which
Supernovae Ia (SNIa) emit radiation (see Perlmutter et al. 1999;
Riess et al. 2001) and of the physical processes that produced
temperature fluctuations in the primeval plasma (see Lee et al.
Centre de Physique Théorique is UMR 6207 - “Unité Mixte de
Recherche” of CNRS and of the Universities “de Provence”, “de
la Méditerranée” and “du Sud Toulon-Var”- Laboratory affiliated to
FRUMAM (FR 2291).
2001; de Bernardis et al. 2002; Halverson et al. 2002; Spergel
et al. 2006.)
Even if the ambitious task of determining geometry and evolution of the universe as a whole, which commenced in the
1930s, now-day shows that the relativistic Friedman-Lemaître
model passes impressively demanding checks, we are faced with
the challenge of developing and adding new lines of evidence
supporting (or falsifying) the concordance model. Moreover,
even if we parameterize our ignorance about dark energy describing its nature only via a simple equation of state w = p/ρ,
we only have loose constraints on the precise value of the w parameter or on its functional behavior.
In this spirit we focus this analysis on possible complementary approaches to determining fundamental cosmological parameters, specifically on geometrical tests.
A whole arsenal of classical geometrical methods has been
developed to measure global properties of the universe. The central feature of all these tests is the attempt to directly probe the
various operative definitions of relativistic distances by means
of scaling relationships in which an observable is expressed as
a function of redshift (z) and of the fraction of critical density
contributed by all forms of matter and energy (Ω).
The most remarkable among these classical methods are
the Hubble diagram (or magnitude-redshift relation m =
m(M, z, Ω)), the Angular diameter test (or angle-redshift rela-
2
C. Marinoni et al.: Geometrical Tests of Cosmological Models with High-z Galaxies. I.
tion θ = θ(L, z, Ω)), the Hubble test (or count-redshift relation
N = N(n, z, Ω)) or the Alcock-Paczinsky test (or deformationredshift relation Δz/zΔθ ≡ k = k(z, Ω)). The common key idea
is to constrain cosmological parameters by measuring, at various cosmic epochs, the scaling of the apparent values m, θ, N,
k of some reference standard in luminosity (M), size (L), density (n) or sphericity and compare them to corresponding model
predictions.
The observational viability of these theoretical strategies has
been remarkably proved by the Supernova Cosmology Project
(Perlmutter et al. 1999) and the High-z Supernova Team (Riess
et al. 2001) in the case of the Hubble diagram. With a parallel
strategy, Newman et al. (2002) recently showed that a variant of
the Hubble test (N(z) test) can be in principle applied to distant
optical clusters selected in deep redshift survey such as VVDS
(Le Fèvre et al. 2005) and DEEP2 (Davis et al. 2000), in order
to measure the cosmic equation-of-state parameter w.
Unfortunately, the conceptually simple pure geometrical
tests of world models, devised to anchor relativistic cosmology
to an observational basis, have so far proved to be difficult to implement. This is because the most effective way to constrain the
evolution of the cosmological metric consists in probing deep
regions of the universe with a primordial class of cosmological
objects. Besides the complex instrumental technology this kind
of experiments requires, it becomes difficult at high redshift to
disentangle the effects of object evolution from the signature of
geometric evolution.
Since geometrical tests are by definition independent from
predictions of theoretical models or simulations, as well as from
assumptions about content, quality and distribution of matter
in the universe (mass fluctuations statistics, e.g. Haiman et al.
2001; Newman et al. 2001, galactic biasing, e.g. Marinoni et al.
1998; Lahav et al. 2001; Marinoni et al. 2006, Halo occupation models, e.g. Berlind et al. 2001; Marinoni & Hudson 2002;
van den Bosch et al. 2006) it is of paramount importance to try
to devise an observational way to implement them. The technical maturity of the new generation of large telescopes, multi
object spectrographs, large imaging detectors and space based
astronomical observatories will allow these tests to be more effectively applied in the near future (Huterer & Turner 2000). In
this paper, we describe a method to select a class of homologous galaxies that are at the same time standard in luminosity
and size, that can be in principle applied to data coming from
the zCOSMOS spectro-photometric survey (Lilly et al. 2006);
of the deep universe.
An observable relationship exists between the speed of rotation V of a spiral galaxy and its metric radial dimension D as
well as its total luminosity L (Tully & Fisher 1977; Bottinelli
et al. 1980). From a theoretical perspective, this set of scaling
relations are expected and explicitly predicted in the context of
CDM models of galaxy formation (Mo et al. 1998). The TullyFisher relations for diameter and luminosity have been extensively used in the local universe to determine the distances to
galaxies and the value of the Hubble constant. We here suggest
that they may be used in a cosmological context to select in a
physically justified way, high redshift standard rods since galaxies having the same rotational speed will statistically have the
same narrow distribution in physical sizes.
The picture gets complicated by the fact that the standard
model of the universe implies some sort of evolution in its constituents. In a non static, expanding universe, where the scale
factor changes with time, we expect various galaxy properties,
such as galaxy metric dimensions, to be an explicit function of
redshift. In principle, one may break this circular argument be-
tween model and evolution with two strategies: either by understanding the effects of different standard rod evolutionary patterns on cosmological parameters, or by looking for cosmological predictions that are independent from the specific form of the
disc evolution function.
In this paper, following the first approach, we study how different disc evolution functions may bias the angular-diameter
test. We simulate the diameter-redshift experiment using the
amount of data and the realistic errors expected in the context
of the zCOSMOS survey. We then evaluate how different disc
evolution functions may be unambiguously recognized from the
data and to what extent they affect the estimated values of the
various cosmological parameters. We also explore the second
approach and show how cosmological information may be extracted, without any knowledge about the particular functional
form of the standard rod evolution, only by requiring as a prior
an estimate of the upper limit value for the relative disc evolution
at some reference redshift.
This paper is set out as follow: in Sect. 2 we review the theoretical basis of the angular diameter test. In Sect. 3 we describe
the proposed strategy to select high redshift standard rods. In
Sect. 4 we digress on how implement in practice the “θ − z” test
with zCOSMOS data, and in Sect. 5 we present the zCOSMOS
expected statistical constraints on cosmological parameters. In
Sect. 6 we discuss different possible approaches with which to
address the problem of standard rod evolution. Conclusions are
drawn in Sect. 7.
2. The angular diameter test
We investigate the possibility of probing the cosmological metric using the redshift dependence of the apparent angular diameter of a cosmic standard rod. What gives this test special appeal
is the possibility of detecting the “cosmological lensing” effect,
which causes incremental magnification of the apparent diameter of a fixed reference length.
Let’s consider the transverse comoving distance (see Hogg
1999)
z
c
r(z, Ωm , ΩQ , w) =
S k | Ωk |
E(x)−1 dx
(1)
√
H0 | Ω k |
0
where
1/2
E(x) = Ωm (1 + x)3 + ΩQ (1 + x)3+3w + Ωk (1 + x)2
(2)
and where S −1 (y) = sinh(y), S 1 (y) = sin(y), S 0 (y) = y while
Ωk = 1 − Ωm − Ω Q .
An object with linear dimension D at a redshift z has thus an
observed angular diameter θ
θ(z, p) =
D
(1 + z)
r(z, p)
(3)
which depends on the general set of cosmological parameters
p = [Ωm , ΩQ , w] via the relativistic definition of angular distance, dA = r(z, p)/(1 + z).
This test may be implemented without requiring the knowledge of the present expansion rate of the universe (the dependence from the Hubble constant cancels out in Eq. (3)). At variance, although characterized by a smooth and diffuse nature,
dark energy significantly affect the dynamic of the universe.
From Eq. (1) it is clear that the angular-diameter test depends on
the dark energy component via the expansion rate evolution E(z).
C. Marinoni et al.: Geometrical Tests of Cosmological Models with High-z Galaxies. I.
3
ogy, and specifically from the angular diameter test, by devising
observational programs probing a large field of view in the redshift range 0 ≤ z ≤ 2.
3. The standard rod
Fig. 1. The relative sensitivity of the angular diameter distance (dA ) and
volume element (dV/dz) to a change in the values of Ωm , ΩQ and w.
The partial derivatives are computed with respect to the position (Ωm =
0.3, ΩQ = 0.7, w = −1) in the parameter space.
The more negative w, the more accelerated the expansion is and
the smaller a fixed standard rod will appear to an observer.
The efficiency of different cosmological observables in probing the nature of space-time ultimately depends upon their sensitivity to the cosmological parameters Ωm , ΩQ , w. The relative
sensitivity of empirical cosmological tests based on the scaling
of the angular diameter distance (dA ) and of the volume element
(dV/dz = (c/H0 )(r2 /E(z)) is derived in Fig. 1, where we assume that Poissonian errors are constant in time and no redshiftdependent systematics perturb the measurements (e.g. Huterer
& Turner 2000). Since the luminosity distance (i.e. the distance
inferred from measurements of the apparent magnitude of an object of known absolute luminosity) is defined as dL = (1 + z)2dA ,
we note that the angular diameter test has the same cosmological
discriminatory power as the Hubble diagram. The upper panel
of Fig. 1 shows that the sensitivity of both dA and dV/dz to the
mean mass density parameter, Ωm , increases monotonically as
a function of redshift. This means that the deeper the region of
the universe surveyed, the more constrained the inferred value
of Ωm is.
Conversely, the sensitivity of both empirical tests to a change
in the constant value of w peaks at redshift around unity, and
levels off at redshifts greater than ∼5. The reason for this is that
the dark energy density ρQ , which substantially contributes to
the present-day value of the expansion rate was negligible in the
early universe (ρQ /ρ M ∝ (1 + z)3w , see Eq. (2)).
The fact that we are living in a special epoch, when two
or more terms in the expansion rate equation make comparable contributions to the present value of E(z), can be appreciated in the central panel of Fig. 1. Because each of the terms
in Eq. (2) varies with cosmic time in a different way, there is
a redshift window where the search for ΩQ is less efficient (i.e.
2 < z < 4). Therefore one can maximize the cosmological information which can be extracted from the classical tests of cosmol-
A variety of standard rod candidates have been explored in previous attempts of implementing the angular diameter-redshift test:
galaxies (Sandage 1972; Djorgovski & Spinrad 1981), clusters
(Hickson 1977; Bruzual & Spinrad 1978; Pen 1997), halo clustering (Cooray et al. 2001). Those methods failed to yield conclusive evidence because the available redshifts were few and
local, and the quality of the imaging data used in the estimate of
sizes was poor.
Good quality size measurements for high redshift objects
have become available for radio sources (e.g. Miley 1971;
Kapahi 1975; and recently several authors Kellermann 1993;
Wilkinson et al. 1998, have reported a redshift dependence of
radio source angular sizes at 0.5 < z < 3, which is not easily
reconciled with other recent measurements of the cosmological
parameters (but see Daly & Djorgovski 2004, for results more
consistent with the concordance model.)
The radio source results may be affected by a variety of selection and evolutionary effects, the lack of a robust definition
of size, and by difficulties in assembling a large, homogeneous
sample of radio observations (Buchalter et al. 1998; Gurvits et al.
1999).
A common thread of weakness in all these studies is that
there are no clear criteria by which galaxies, clusters, extended
radio lobes or compact radio jets associated with quasars and
AGNs should be considered universal standard rods. Moreover,
lacking any local calibration for the metric size of the standard
rod, the standard rod dimension (parameter D in Eq. (3)) is often
considered as a free fitting parameter. Since the inferred cosmological parameters heavily depend on the assumed value for
the object size (Lima & Alcaniz 2002), an a priori independent
statistical study of the standard rod distribution properties is an
imperative prerequisite.
We thus propose to use information on the kinematics of
galaxies, as encoded in their optical spectrum, a) to identify in an
objective and empirically justified way a class of objects behaving as standard rods, and b) to measure the absolute value of the
standard rod length. The basic idea consists in using the velocitydiameter relationship for disc galaxies (e.g. Tully & Fisher 1977;
Saintonge et al. 2007) as a cosmological metric probe. In Fig.
2 we plot the local relationship we have derived in Paper II of
this series (Saintonge et al. 2007) between half-light diameters
and rotational velocities inferred using the Hα λ6563 Å line. The
sample used to calibrate the diameter-velocity relationship is the
SFI++ sample described by Springob et al. (2007). Also shown
is the the amplitude of the scatter in the zero-point calibration of
the standard rods.
In the local universe, rotation velocities can be estimated
from either 21 cm HI spectra or from the Hα λ6563 Å optical
emission lines. However, the Hα line is quickly redshifted into
the near-infrared and cannot be used in ground-based optical
galaxy redshift surveys at z > 0.4, while HI is not detectable
much past z = 0.15. Only [OII]λ3727 Å line widths can be successfully used in optical surveys to infer the length of a standard
rod dimension D at z ∼ 1. Clearly one could obtain rotational
velocity information for high redshift objets by observing the
Hα line with near-IR spectrographs. However it is much easier
to get large samples of kinematic measurements using OII and
4
C. Marinoni et al.: Geometrical Tests of Cosmological Models with High-z Galaxies. I.
4. Optimal test strategies
In this section we outline the optimal observational strategy
required in order to perform the proposed test. With the proposed selection technique, the photometric standard rod D is
spectroscopically selected and the sample is therefore free from
luminosity-size selection effects, that is from the well known
tendency to select brighter and bigger objects at higher redshifts
(Malmquist bias) in flux-limited samples. However it is crucial
√
that a large sample of spectra be collected, in order to obtain N
gain over the intrinsic scatter in the calibrated V(OII)-diameter
relationship.
4.1. Galaxy sizes measurement
Fig. 2. The diameter vs. velocity relationship calibrated in Paper II is
plotted using a black line. Shadowed regions represent the 1 and 2σ
uncertainties in the calibrated relationship. D represents the corrected
(face on) half light diameter while the velocity has been measured using the Hα line. Dotted lines represent the upper and lower relative
uncertainty [σD /D]int = 0.15 in diameters. The conservative relative
dispersion in the relationship assumed in this study ([σD /D]int = 0.4)
is also presented using dashed lines. With this conservative choice
we take into account that sizes and velocity are measured in the high
redshift universe with greater uncertainties. Within the interval centered at V = 200 ± 20 km s−1 (mean physical galaxy dimension
∼10 ± 1.5 h−1 kpc which roughly corresponds to an isophotal diameter D25 ∼ 20 ± 3 h−1 kpc), we will select galaxies with total absolute
I magnitude MI − 5 log h = −22.4 (see Paper II). The velocity selected
standard rods at z ∼ 1 are thus well within the visibility window of
deep galaxy redshift surveys such as the VVDS (Le Fèvre et al. 2005)
or zCOSMOS (Lilly et al. 2006). For exemple the VVDS is flux limited
at I < 24 and selects objects brighter then MI ∼ −20 + 5 log h at z = 1.
multi-slit devices, rather than get sparser samples using a singleobject, near-IR spectrograph.
A detailed study of the kinematical information encoded in
the [OII] line are presented in Paper II. In that paper we have explored the degree of correlation of optical Hα and [OII] rotational
velocity indicators, i.e. how well the rotation velocities extracted
from these different lines compare. Moreover, we have derived
a local diameter-velocity relationship, and we have investigated
the amplitude of the scatter in the zero-point calibration of the
standard rods.
We finally note that the present-day expansion rate sets the
overall size and time scales for most other observables in cosmology. Thus, if we hope to seriously constrain other cosmological parameters it is of vital importance either to pin down
its value or to devise H0 -independent cosmological tests. Note
that, given the calibration of the diameter–linewidth relation in
the form H0 = f (V), the θ-expression in Eq. (3) is effectively
independent of the value of the Hubble constant.
Since galaxies do not have sharp edges, their angular diameter
is usually defined in terms of isophotal magnitudes. However
since surface brightness is not constant with distance, the success
in performing the experiment revolves around the use of metric
rather than isophotal galaxy diameters (Sandage 1995).
A suitable way to measure the photometric parameter θ,
without making any a-priori assumption about cosmological
models, consists in adopting as the standard scale length estimator either the half-light radius of the galaxy or the η-function
of Petrosian (1976). The Petrosian radius is implicitly defined as
η(θ) =
μ(θ)
,
μ(θ)
(4)
i.e. as the radius θ at which the surface brightness averaged inside θ is a predefined factor η larger than the local surface brightness at θ itself.
Both these size indicators are independent of K-correction,
dust absorption, luminosity evolution (provided the evolutionary
change of surface brightness is independent of radius), waveband used (if there is no color gradient) and source light profile
(Djorgovski & Spinrad 1981).
4.2. Standard rods optimal selection
The choice of the objects for which the velocity parameter V and
the metric size is to be measured is a compromise between the
observational need of detecting high signal-to-noise spectral and
photometric features (i.e. selecting high luminosity and large objects) and the requirement of sampling the velocity distribution
function (n(V)dV ∼ V −4 for galaxy-scale halos) within an interval where the rotator density is substantial.
Given the estimated source of errors (see next section), and
the requirement of determining both ΩQ and w with a precision of 20%, we find, guided by semi analytical models predicting the redshift distribution of rotators (i.e. Narayan & White
1988; Newman & Davis 2000), that an optimal choice are V =
200 km s−1 rotators.
In particular, as shown in Paper II, the I band characteristic absolute magnitude of the V = 200 km s−1 objects is
MI ∼ −22.4 + 5 log h70 i.e. well above the visibility threshold
of flux-limited surveys such as zCOSMOS or VVDS (as an example, for Iab = 24 the limiting magnitude of the VVDS, one
obtains that MI −21 + 5 log h70 at z = 1.5). Therefore, with
this velocity choice, the selected standards do not suffer from
any Malmquist bias (i.e. any effect which favors the systematic
selection of the brighter tails of a luminosity distribution at progressively higher redshifts).
C. Marinoni et al.: Geometrical Tests of Cosmological Models with High-z Galaxies. I.
4.3. The zCOSMOS potential
In Sect. 2 we have emphasized that an optimal strategy to study
the expansion history of the universe consists in probing, with
the angular diameter test, the 0 ≤ z ≤ 2 interval. However, in
order to collect a large and minimally biased sample of standard
rods over such a wide redshift baseline, we need the joint availability of high quality images and high resolution multi-object
spectra.
Space images allow a better determination of galaxy structural parameters (sizes, luminosity, surface brightness, inclination etc) at high redshift. In particular, the ACS camera of the
Hubble Space Telescope can survey large sky regions with the
key advantages of a space-based experiment: diffraction limited
images no seeing blurring, and very deep photometry. Ground
multi-object spectrographs operating in high resolution mode allow a better characterisation of the gravitational potential-well of
galaxies, facilitating the fast acquisition of large samples of standard rods. For example, in the spectroscopic resolution mode
R = 2500 (1
slits), the VIMOS spectrograph (Le Fèvre et al.
2003) allows to resolve the internal kinematics of galaxies via
their rotation curve or line-width. Note that the VIMOS slitlets
can be tilted and aligned along the major axis of the galaxies in
order to remove a source of potentially significant error in the
estimate of the rotation velocities.
These observational requirements are mandatory for a succesfull implementation of the proposed cosmological probe. The
practical feasibility of the strategy is graphically illustrated in
Fig. 3. In this figure we show a high resolution spectra of a
galaxy at z = 0.5016 obtained with a total exposure of 90 min
with VIMOS. The ground and space (ACS) images of the galaxy
are also shown for comparison.
Interestingly, a large sample of rotators can be quickly assembled by the currently underway zCOSMOS deep redshift
survey, which uses the VIMOS multi-object spectrograph at the
VLT to target galaxies with ACS photometry in the 2 sq. degree
COSMOS field Scoville et al. 2003). In principle, one can measure in high resolution modality R = 2500 the line-widths of
[OII]λ3727 Å up to z ∼ 1.4 by re-targeting objects for which
the redshift has already been measured in the low-resolution
(R = 600) zCOSMOS survey. Since the spectral interval covered in high resolution mode is limited to ∼2000 Å, this allows
us to determine the optimal telescope and slitlets position angles
in order to maximize the number of spectra whose OII emission
lines fall onto the CCDs. With this fast follow-up strategy one
could be able to target about 200 rotators per pointing with an
exposure time of 4 h down to Iab = 24.
In the following we will use the zCOSMOS sample as a test
case to assess the performances of the proposed cosmological
test.
5. Constraints on cosmological parameters
In this section we present in detail some merits and advantages
of the proposed approach for constraining the value of the fundamental set of cosmological parameters. We evaluate the potential
of the test, within the zCOSMOS operational specifications, in
placing constraints not only on the simplest models of the universe, which include only matter and a cosmological constant,
but also on so-called “quintessence” models (Turner & White
1997; Newman et al. 2002), For the purposes of this study, we
assume in this section w to be constant in time up to the redshift
investigated z ∼ 1.4.
5
We assume that a V(OII)-diameter relation can be locally
calibrated, and that the diameter of V = 200 km s−1 rotators may
be inferred with a worst(/optimal) case relative error [σD /D]int =
40%(/15%) (see Sect. 3 and Fig. 2). The main contributions to
this error figure are uncertainties in measuring linewidths, galaxy
inclinations and the intrinsic scatter of the empirical relation itself. We then linearly combine this intrinsic scatter with the uncertainties with which the Petrosian or half-light radius may be
determined from the ACS photometry. The error on the half light
radii is about σθ = ±0.04
almost independent of galaxy sizes
in the range 0.1–1
(S. Gwyn, private communication).
We consider as the observable of the experiment the logarithm of the angle subtended by a standard rod. We use the logarithms of the angles rather than angles themselves because we
assume that the object magnitudes, rather than diameters, are
normally distributed around some mean value. Moreover, in this
way the galactic diameter becomes an additive parameter, whose
fitted value (when a z = 0 calibration is not available) does not
distort the cosmological shape of the θ(z) function.
The observed values log θ are randomly simulated around the
theoretical value log θr (cf. Eq. (3)) using the standard deviation
given by
σ 2
σ 2 1/2
D
θ
σ=
+
·
(5)
D int
θ obs
For the purposes of this section, θr is computed assuming a flat,
vacumm dominated cosmology with parameters Ωm = 0.3, ΩQ =
0.7, and w = −1 as reference model. The confidence with which
these parameters are constrained by noisy data is evaluated using
the χ2 statistic
χ2 =
[log θi − log θth (z, p)]2
i
i
σ2
,
(6)
where θth is given by Eq. (3).
We also derive the expected redshift distribution of galaxies
having circular velocity in the 180 ≤ V ≤ 220 km s−1 range,
in various cosmological scenarios within the framework of the
Press & Schechter formalism (Narayan & White 1988; Newman
& Davis 2000). We note that due to the correlation between circular velocity and luminosity, these galaxies could be observed
to the maximum depth (z ∼ 1.4) out to which the OII is within
the visibility window of VIMOS. We take into account the uncertainties in the semi-analytic predictions and our ignorance about
the fraction of discs to be observed that will have a spectroscopically resolved [OII]λ3727 Å line, by multiplying the calculated halo density by a “conservative” factor f = 0.2. Using the
VIMOS R = 2500 resolution mode data, we thus expect to be
able to implement our angular-diameter research program using
nearly 1300 standard rods per square degree, which is what has
been simulated (see Fig. 4).
Since we do not make a-priori assumptions about any parameter, and in particular we do not assume a flat cosmology,
results should be distributed as a χ2ν with ν = 3 degrees of freedom, which can be directly translated into statistical confidence
contours, as presented in Fig. 5. This figure shows that even
without assuming a flat cosmology as a prior and considering
a diameter-linewidth relationship with a 40% scatter, by targetting ∼1300 rotators we can directly infer the presence of a dark
energy component with a confidence level better than 3σ. At
the same time, its equation of state can be constrained to better than 20% (∼10% if the the diameter-linewidth relationship is
calibrated with a ∼15% relative precision).
6
C. Marinoni et al.: Geometrical Tests of Cosmological Models with High-z Galaxies. I.
2 arcsec
Z=0.5016
F775W=19.5
I=39 deg
PA=43 deg
Fig. 3. Upper: public release image of the galaxy α = 53.1874858, δ = −27.910975 at redshift 0.5016, taken with the filter F775W (nearly I band)
of the ACS camera by the GOODs program. For comparison the same galaxy as imaged in the EIS survey with the WFI camera at the ESO 2.2mt
telescope at La Silla. Botton left: raw spectrum of the galaxy taken by VIMOS at the VLT-UT3 telescope with a total exposure time of 90 min
and a spectral resolution R = 2000. Slitlets have been tilted according to the major axis orientation (Position Angle = 43◦ ). Botton right: final
processed spectrum showing the rotation curve as traced by the Hb (λ4859 Å) line.
6. Standard rod evolution
The previous analysis shows that the angular diameter test, when
performed using fast high resolution follow-up of zCOSMOS
spectroscopic targets may be used as a promising additional
tool to explore the cosmological parameter space and directly
measure a dark energy component. However, the impact of any
standard rod evolution on these results needs to be carefully
examined.
First of all, we may note that the expected variation with
cosmic time of the total galaxy luminosity due to evolution in its
stellar component does not affect the metric definition of angular
diameters unless this luminosity change depends on radius (see
Paper III of this series, marinoni et al. 2007, for a detailed analysis of this issue). Moreover, we can check each galaxy spectrum
or image for peculiarities indicating possible evolution or instability of the standard rod which may be induced by environmental effects, interactions or excess of star formation.
Any possible size evolution of the standard rod needs to
be taken into account next. Interestingly, it has been shown
by different authors that large discs in high redshift samples
evolve much less in size than in luminosity in the redshift range
0 < z < 1. Recent studies show that the amount of evolution to
z ∼ 1 appears to be somewhat smaller than expected: disc sizes
at z ∼ 1 are typically only slightly smaller than sizes measured
locally (Takamiya 1999; Faber et al. 2001; Nelson et al. 2002;
Totani et al. 2002). This is also theoretically predicted by simula-
tions; Boissier & Prantzos (2001), for example, show that large
discs (i.e. fast rotators) should have basically completed their
evolution already by z ∼ 1 and undergo very little increase in size
afterwards. Infall models (e.g., Chiappini et al. 1997; Ferguson
& Clarke 2001; Bouwens & Silk 2002) also predict a mild disk
size evolution. Disk sizes at z ∼ 1 in these models are typically
only 20% smaller than at z = 0.
6.1. Analysis of the biases introduced by evolution
Even if literature evidences are encouraging, we have to be
aware that even a small amount of evolution may introduce
artificial features and bias the reliability of the cosmological
inferences.
In this section we directly address this issue by considering
different evolutionary patterns for the standard rods, and by analyzing to what level the simulated true cosmological model may
still be correctly inferred using evolved data. In other words, we
investigate how different disc evolutionary histories affect the
determination of cosmological parameters by answering to the
following three questions:
a) is there a feature that may be used to discriminate the presence of evolution in the data?
b) which cosmological parameter is more sensitive to the eventual presence of disc evolution?, and in particular what are the
C. Marinoni et al.: Geometrical Tests of Cosmological Models with High-z Galaxies. I.
7
Fig. 4. Left: redshift evolution of the differential comoving number density of halos with a circular velocity of 200 km s−1 computed according to
the prescriptions of Newman & Davis (2000) in the case of a flat cosmological model having Ωm = 0.3 today and a cosmological constant. Only
a fraction f = 0.2 of the total predicted abundance of halos (i.e. ∼1300 objects per square degree) is conservatively supposed to give line-width
information useful for the angular-diameter test. Right: simulation of the predicted scatter expected to affect the angular diameter-redshift diagram
should in principle achieve with the angular diameter test. The simulation is performed assuming the sample is composed by ∼1300 rotators with
V = 200 km s−1 and that a flat model with parameters [Ωm = 0.3, ΩQ = 0.7, w = −1] is the true underlying cosmological framework. The circular
velocity has been converted into an estimate of the galaxy diameter (Dv = 20 kpc) by using the velocity-diameter template calibrated by Bottinelli
et al. (1980). The worst-case scenario ([σD /D]int = 40%) is presented. The solid line visualizes the underlying input cosmological model θΛCDM (z),
while triangles are drawn from the expected Poissonian fluctuations. The dot-dashed line represents the expected scaling of the angular diameter
in our best recovered cosmological solution. The dashed and the dotted lines represent the angular scaling in a Einstein-de Sitter and Euclidean
(non-expanding type cosmology with zero curvature) geometry respectively.
effects of evolution on the value of the inferred dark energy
density parameter ΩQ ?
c) is there a particular evolutionary scenario for which the inferred values of ΩQ and w are minimally biased?
For the purposes of this analysis, we consider for the angular
diameter-redshift test the baseline (0.1 ≤ z ≤ 1.4) divided in bins
of width dz = 0.1 and assume a relative scatter in the mean size
per bin of 5%. This scatter nearly corresponds to that expected
for a sample of 1300 rotators (with 0 < z < 1.4 and dz = 0.1)
whose diameters are individually (and locally) calibrated with
a 40% precision. We then select a given fiducial cosmology
(input cosmology), apply an arbitrary evolution to the standard
rods, and then fit the evolved data with the unevolved theoretical prediction given in Eq. (3) in order to obtain the best fitting
(biased) output cosmology and the associated confidence levels
contours. We decide that the best fitting cosmological model offers a good fit to the evolved data if the probability of a worse χ2
is smaller that 5% (i.e. P(χ2 > χ2obs ) < 0.05).
We adopt three different parameterizations to describe an
eventual redshift evolution of the velocity selected sample of
galaxy discs Dv : a late-epoch √
evolutionary scenario (ΔD/D ≡
(Dv (z) − Dv (0))/Dv(0) = −|δ1 | z) where most of the evolution
is expected to happen at low redshifts and levels off at greater
distances (δ1 is the relative disc evolution at z = 1), a linear evolutionary scenario (ΔD/D = −|δ1 |z) without any preferred scale
where major evolutionary phenomena take place (i.e. the gradient of the evolution is nearly constant), and an early-epoch evolution scenario (ΔD/D = −|δ1 |z2 ) where most of the evolution is
expected to happen at high redshift.
We note that for modest disc evolution, the linear parameterization satisfactorily describes the whole class of evolutionary models whose series expansion may be linearly represented
(for example, the hyperbolic model (Dv (z) = Dv (0)/(1 + |δ1 |z))).
For z 1 it also represents fairly well the exponential model
(Dv (z) = Dv (0)(1−z)δ). Moreover, the linear model is the favored
scenario for disc size evolution at least at low redshift (1.5) as
predicted by simulations (e.g. Mo et al. 1998; Bouwens & Silk
2002).
First, let’s assume that w = −1 and that the dark energy behaves like Einstein’s cosmological constant. In Fig. 6 we consider three different input fiducial cosmological models (a flat
Λ-dominated universe (ΩΛ = 0.7), an open model (Ωm = 0.3)
and an Einstein-de Sitter universe (Ωm = 1)) and show the characteristic pattern traced by the best fitting output values (Ωm ,
8
C. Marinoni et al.: Geometrical Tests of Cosmological Models with High-z Galaxies. I.
Cosmological Constraints
Fig. 5. Predicted 1, 2, and 3σ confidence level contours for application of the angular-diameter test. The likelihood contours have been derived by
adopting a Λ-cosmology [Ωm , ΩQ , w] = [0.3, 0.7, −1] as the fiducial model we want to recover, and by conservatively assuming one can obtain
useful line-widths information for a sample of galaxies having the redshift distribution and the Poissonian diameter fluctuations simulated in
Fig. 4. Confidence contours are projected onto various 2D-planes of the [Ωm , ΩQ , w] parameter space, and the jointly best fitted value along the
projection axis, together with the statistical significance of the fit, are reported in the insets. Note the strong complementarity of the confidence
region orientation which is orthogonal to the degeneration axis of the CMB measurements. Top: constraints derived assuming that one might
survey only 1 square degree of sky and that the V(OII)-diameter relation is locally calibrated with a 40% of relative scatter in diameter. Center:
as before but assuming a scatter of 15% in diameters. Bottom: confidence contours for a survey of 16 deg2 (which corresponds to the full area
surveyed by VIMOS-VLT Deep Survey) assuming a template V(OII)-diameter relationship with a scatter of 30% in diameters.
ΩΛ ) inferred by applying the angular diameter test to data affected by evolution. A common feature of all the various evolutionary schemes considered is that the value of Ωm is systematically underestimated with respect to its true input value: the
stronger the evolution in diameter and the smaller Ωm will be,
irrespectively of the particular disc evolutionary model considered. Since many independent observations consistently indicate
the existence of a lower bound for the value of the normalized
matter density (Ωm 0.2), we can thus use this parameter as a
sensitive indicator of evolution.
energy determination. If the gradient of the disk evolution function increases with redshift (quadratic evolution), then the estimates of ΩΛ are systematically biased low. The contrary happens
if the evolutionary gradient decreases as a function of look-back
time (square root model). If the disc evolution rate is constant
(linear model), then even if discs are smaller by a factor as large
as 40% at z = 1.5 the estimate of the dark energy parameter is
only minimally biased. The net effect of a linear evolution is to
approximately shift the best fitting ΩΛ value in a direction parallel to the Ωm axis in the [Ωm , ΩΛ ] plane.
Once the presence of evolution is recognized, the remaining
problem is to determine the level of bias introduced in the dark
More generally, by linearly evolving disc sizes so that they
are up to 40% smaller at z = 1.5, and by simulating the apparent
C. Marinoni et al.: Geometrical Tests of Cosmological Models with High-z Galaxies. I.
9
Fig. 6. Best fitting cosmological parameters inferred by applying the angular diameter test to data affected by evolution. The output (biased)
estimates of Ωm and ΩΛ are plotted as a function of the relative diameter evolution for the following evolutionary models: linear (solid line),
square-root (dotted line) and quadratic evolution (dashed line). The biasing pattern is evaluated for three different fiducial cosmologies: a flat,
Λ-dominated cosmology (ΩΛ = 0.7, left), a low-density open cosmology (Ωm = 0.3, center) and a flat, matter-dominated model (Ωm = 1, right).
angle observed in any arbitrary cosmological model with matter
and energy density parameters in the range 0 < Ωm < 1 and
0 ≤ ΩΛ ≤ 1 (w = −1) we conclude that the maximum deviation
of the inferred biased value of ΩΛ from its true input value, is
limited to be |max(δΩΛ )| 0.2, whatever the true input value of
the energy density parameter is. In other terms, in the particular
case of a linear and substantial (<40% at z = 1.5) evolution of
galaxy discs, the central value of the dark energy parameter is
minimally biased for any fiducial input model with 0 ≤ Ωm < 1
and 0 ≤ ΩΛ ≤ 1).
We have shown that the presence of evolution is unambiguously indicated by the “unphysical” best fitting value of the parameter Ωm . We now investigate the amplitude of the biases induced by disc evolution in the [ΩQ , w] plane. We assume for this
purpose that w is free to vary in the range −1 ≤ w ≤ −1/3, which
means assuming that the late epoch acceleration of the universe
might be explained in terms of a slow rolling scalar field.
We first consider a situation where the disc size evolution
is modest, and could be represented by any of the three models
considered. Whatever the mild evolution model considered (less
then 15% evolution from z = 1.5) and assuming a scatter in the
angular diameter-redshift diagram of 5% in each redshift bin,
we find that the input values of ΩQ and w are contained within
the 1σ biased confidence contour derived from the evolved data.
Figure 7 shows the 1, 2 and 3σ “biased” confidence contours
obtained by fitting with Eq. (3) a simulated angular-diameter
redshift diagram in which discs have been linearly evolved. We
show that, if disc evolution depends linearly on redshift and
causes galaxy dimensions to be up to 30% smaller at z = 1.5,
the true fiducial input values of ΩQ and w are still within 1σ of
the biased confidence contours inferred in presence of a standard
rod evolution (and a scatter in the angular diameter-redshift relation as low as 5% in each redshift bin). We have tested that
these conclusions hold true for every fiducial input cosmology
10
C. Marinoni et al.: Geometrical Tests of Cosmological Models with High-z Galaxies. I.
Fig. 7. 1, 2, and 3σ confidence level contours in the [ΩQ , w] plane computed by applying the angular-diameter test to data unaffected (upper panel)
and affected by diameter evolution (lower panel). We consider a linear model for disc evolution normalized by assuming that discs were smaller
by 30% at z = 1.5, and a nominal relative error in the standard rod measures of 5% per redshift bin (see discussion in Sect. 6.1). The effects of disc
evolution onto cosmological parameter estimation are compared to the evolution-free case for three different fiducial cosmologies: a low matter
density open cosmology (Ωm = 0.3, left) a flat, Λ-dominated cosmology (ΩΛ = 0.7, center), and an Einstein-de Sitter model (Ωm = 1, right).
with parameters in the range 0 ≤ Ωm ≤ 1, 0 ≤ ΩQ ≤ 1 and
−1 ≤ w ≤ −1/3.
Thus, if, disc evolution is linear (as predicted by theoretical
models) and substantial (up to ∼30% at z = 1.5), or arbitrary
and mild (up to ∼15% at z = 1.5), then in both cases the angular diameter test reduces from a test of the whole set of cosmological parameters, to a direct and fully geometrical test of
the parameters subset (ΩQ , w). For example, in a minimal approach, the angular diameter test could be used to test in a purely
geometrical way the null hypothesis that “a dark energy component with a constant equation of state parameter w is dominating
the present day dynamics of the universe”. Moreover, as Fig. 7
shows, an universe dominated by dark energy may be satisfactorily discriminated from a matter dominated universe (ΩQ = 0).
In the evolutionary pictures considered, galaxy discs are supposed to decrease monotonically in size in the past. Since the
sensitivity of the test to changes in the linear diameter D is described by a growing monotonic function in the redshift interval
[0, 1.4], one may hope to test cosmology in a way which is less
dependent on systematic biases by limiting the sample at z ≤ 1.
However, by doing this we would halve the number of standard
rods available for the analysis (∼600 with respect to the original
∼1300). The test efficiency in constraining cosmological parameters would consequently be significantly degraded.
6.2. The Hubble-diagram using galaxies
The same velocity criterion that allows the selection of standard rods also allows the selection of a sample of standard candles. Using the same set of tracers, the Tully & Fisher relationships connecting galactic rotation velocities to luminosities and
sizes offer the interesting possibility of implementing two different cosmological tests, the angular-diameter test and the Hubble
diagram.
Thus, we have analyzed how the effects of luminosity evolution may bias the estimation of cosmological parameters in the
case of the Hubble diagram. Assuming that the absolute luminosity L of the standard candle increases as a function of redshift according to the square root, linear and quadratic scenarios, then the value of Ωm is systematically overestimated. The
value of Ωm is biased in an opposite sense with respect to the
angular diameter test (see Fig. 6). Therefore, it is less straightforward to discriminate the eventual presence of evolution in the
standard candle on the basis of the simple requirement that any
“physical” matter density parameter is characterized by a positive lower bound. The different Ωm shifts (with respect to the
fiducial value) observed when the evolved data are fitted using
the Hubble diagram or the angualr diameter test are due to the
fact that, given the observed magnitudes and apparent angles, an
increase with redshift in the standard candle absolute luminos-
6.3. The cosmology-evolution diagram
In this section we want to address the more general question of
whether it is possible to infer cosmological information knowing
a-priori only the upper limit value for disc evolution at some
reference redshift (for example, the maximum redshift surveyed
by a given sample of rotators). In other terms, we explore the
possibility of probing geometrically the cosmological parameter
space in a way which is independent of the specific evolution
function with which disc sizes change as a function of time. The
only external prior is the knowledge of an upper limit for the
amplitude of disc evolution at some past epoch.
Given an arbitrary model specified by a set of cosmological
parameters p, and given the observable θobs (z), i.e. the apparent
angle subtended by a sample of velocity selected galaxies (with
locally calibrated diameters Dv (0)), then
θ (z, p) = θobs (z) − θth (Dv (0), z, p)
(7)
is the function which describes the redshift evolution of the standard rod, i.e θ = (Dv (z) − Dv (0))/dθ in the selected cosmology.
Let’s suppose that we know the lower and upper limits of the
v (z̄)|
relative (adimensional) standard rod evolution (δ(z̄) = |ΔD
Dv (0) ) at
some specific redshift z̄.
Assuming this prior, we can solve for the set of cosmological
parameters (i.e. points p of the cosmological parameter space)
which satisfy the condition
δl (z̄) ≤
|θ ( p, z̄)|
≤ δu (z̄).
θth (D(0), z̄, p)
(8)
This inequality establishes a mapping between cosmology and
the amount of disc evolution at a given redshift which is compatible with the observed data. By solving it, one can construct
a self-consistent cosmology-evolution plane where to any given
range of disc size evolution at z̄ corresponds in a unique way a
specific region of the cosmological parameter space. Vice versa,
for any given cosmology one can extract information about disc
evolution. Clearly, the scatter in the angular-diameter diagram
directly translates in the uncertainties associated to the evolution
boundaries in the cosmological space.
< 0.2
5
< 0.2
0
0.20
<δ
0.15
<δ
ity causes the best fitting distances to be biased towards higher
values, while a decrement in the physical size implies that real
cosmological distances are underestimated.
Moreover, even considering a linear evolutionary picture for
the absolute luminosity as well as a modest change in the standard candle luminosity, i.e. Δ M = M(z)− M(0) = −0.5 at z = 1.5,
the input fiducial cosmology falls outside the 3σ confidence contour obtained by applying the magnitude-redshift test to the sample of evolved standard candles. Note that error contours are derived by assuming a scatter of σ M = 0.05 per redshift bin in the
Hubble diagram. Since, at variance with the size of large discs,
galactic luminosity is expected to evolve substantially with redshift (within the VVDS survey, Δ M ∼ −1 in the I band for M∗
galaxies Ilbert et al. 2005) we conclude that the direct implementation of the Hubble diagram test as a minimial test for the parameter subset [ΩQ , w] using galaxy rotation as the standard candle indicator is more problematic. As an additional problem, we
note that galaxy luminosity is seriously affected by uncertainties
in internal absorption corrections, and that due to K-correction,
the implementation of the Hubble diagram requires multi band
images to properly describe the rest frame emission properties
of galaxies.
11
−1.9
3<Δ
Μ<
−1.8
−1.4
0.25
2
4<Δ
<δ<
Μ<
0.30
−
1.30
−1.3
0<Δ
0.30
Μ<
<δ<
−1.1
0.35
5
−1.1
5<Δ
Μ<
0.35
−0.9
<δ<
9
0.40
−0.9
9<Δ
Μ<
−0.8
0.40
2
<δ<
0.45
−0.8
2<Δ
Μ<
−0.6
3
C. Marinoni et al.: Geometrical Tests of Cosmological Models with High-z Galaxies. I.
Fig. 8. Cosmology-evolution diagram for simulated data which are affected by evolution. Apparent angles and luminosities of the velocity selected sample of rotators are simulated in a ΛCDM cosmology.
Standard rods and candles have been artificially evolved so that at
z̄ = 1.5 discs are 26% smaller and luminosities 1.4 mag brigther. The
cosmological plane is partitioned with different boundaries obtained by
solving equation 8 for different values of δ(z̄ = 1.5), i.e. of the external
prior representing the guessed upper limit of the relative disc evolution at the maximum redshift covered by data. The external prior is also
expressed in term of absolute luminosity evolution (see discussion in
Appendix A).
We note that the boundaries of the region of the cosmological parameter space which is compatible with the assumed prior
on the evolution of diameters at z̄ can be equivalently expressed
in term of the maximum absolute evolution in luminosity (see
Appendix A). This because, as stated in the previous section, velocity selected objects have the unique property of being at the
same time standards of reference both in size and luminosity.
With this approach, one may by-pass the lack of knowledge
about of the particular evolutionary track of disc scalelengths
and luminosities and try to extract information about cosmology/evolution by giving as a prior only the fractional evolution
in diameters or the absolute evolution in magnitude expected at
a given redshift. The essence of the method is as follows: instead
of directly putting constraints in the cosmological parameters
space by mean of cosmological probes, we study how bounded
regions in the evolutionary plane ( ΔD
D , ΔM) map onto the cosmological parameter space.
In Fig. 8 we show the cosmology-evolution diagram derived by solving Eq. (8) for different ranges of δ(z̄). The reference model is the concordanace model (Ωm = 0.3, ΩQ = 0.7,
w = −1) and the reference evolution at z̄ = 1.5 is assumed to be
δ(z̄) = 0.25 for discs, and Mv (z̄) − Mv (0) = −1.5 for luminosities.
The cosmology-evolution diagram represents the unique correspondance between all the possible cosmological models and the
amount of evolution in size and luminosity which is compatible
with the observed data at the given reference redshift. Using independent informations about the range of evolution expected
in the structural parameters of galaxies, for exemple from simulations or theoretical models, one may constrain the value of
.16
−1.31 <
ΔΜ < −
1
0.30
ΔΜ < −
0.25 < δ
<
−1.45 <
< 0.25
0.20 < δ
< 0.20
0.15 < δ
0.0 < δ
< 0.05
0.05 < δ
< 0.10
0.10 < δ
< 0.15
1.31
C. Marinoni et al.: Geometrical Tests of Cosmological Models with High-z Galaxies. I.
−1.93 <
ΔΜ < −
1.82
−1.82 <
ΔΜ < −
1.71
−1.71 <
ΔΜ < −
1.58
−1.58 <
ΔΜ < −
1.45
12
Fig. 9. As in Fig. 8 but for a different cosmology (an Einstein-de Sitter
universe).
cosmological parameters Vice versa, if the cosmological model
is known, then one may directly determine the evolution in magnitude and size of the velocity selected sample of rotators.
With this approach, the possibility of discriminating between
different cosmologies depends on the amount of evolution affecting the standard rods at redshift z̄. Two different sets ( p1 and p2 )
of cosmological parameters may be discriminated at z = z̄ if the
relative disc evolution at z̄ is known to a precision better than
δ(z̄) <
|r(z̄, p2 ) − r(z̄, p1 )|
·
r(z̄, p1 )
(9)
For example, this kind of analysis shows that an Einstein-de
Sitter universe may be unambigously discriminated from a critical universe with parameter Ωm = 0.3, ΩQ = 0.7, w = −1. if
the relative disc evolution at z̄ = 1.5 is known with a precision
better than 28%. In a similar way, an open (Ωm = 0.3, ΩQ = 0)
universe may be discriminated from an Einstein de-Sitter universe if δ(z̄) < 17%.
7. Conclusions
The scaling of the apparent angular diameter of galaxies with
redshift θ(z) is a powerful discriminator of cosmological models.
The goal of this paper is to explore the potentiality of a new
observational implementation of the classical angular-diameter
test and to study its performances and limitations.
We propose to use the velocity-diameter relationship, calibrated using the [OII]λ3727 Å line-widths, as a tool to select
standard rods and probe world models. As for other purely geometrical test of cosmology, a fair sampling of the galaxy population is not required. It is however imperative to have high
quality measurements of the structural parameters of high redshift galaxies (disc sizes and rotational velocity). Surveys with
HST imaging and high enough spectral resolution will thus provide the fundamental ingredients for the practical realization of
the recipe we have presented.
In order to avoid any luminosity dependent selection effect
(such as for exemple Malmquist bias) it is necessary to apply the proposed test to high velocity rotators. We show that
nearly 1300 standard rods with rotational velocity in the bin
V ∼ 200 ± 20 km s−1 ) are expected in a field of size 1 deg2 over
the redshift range 0 < z < 1.4. Interestingly this large sample
can be quickly assembled by the currently underway zCOSMOS
deep redshift survey, which uses the VIMOS multi-object spectrograph at the VLT to target galaxies photometrically selected
using high-resolution ACS images.
Even allowing a scatter of 40% for the
[OII]λ3727Å linewidth-diameter relationship for disc galaxies,
we show that the angular-diameter diagram constructed using
this sample is affected by a scatter of only ∼5% per redshift
bin of amplitude dz = 0.1. This scatter translates into a 20%
precision in the “geometric” measurement of the dark energy
constant equation of state parameter w, through a test performed
without priors in the [Ωm , ΩQ ] space.
Current theoretical models suggest that large discs (i.e. fast
rotators) evolve weakly with cosmic time from z = 1.5 down
to the present epoch. Anyway, we have explored how an eventual evolution of the velocity-selected standard rods might affect
the implementation of the test. We have shown that any possible
evolution in the standard rods may be unanbiguously revealed
by the fact that even a small decrement with redshift of the disc
sizes shifts the inferred value of the matter density parameter
into “a-priori excluded” regions (Ωm < 0.2).
We have shown that a linear (as expected on the basis of
various theoretical models) and substantial (up to 40% over the
range 0 < z < 1.5) disc evolution minimally biases the inferred
value of a dark energy component that behaves like Einstein’s
cosmological constant Λ. Moreover we have shown that assuming that discs evolve in a linear-like way as a function of redshift,
and that their sizes were not more than 30% smaller at z = 1.5
with respect to their present epoch dimension, then the angular
diameter test can be used to place interesting constraints in the
[ΩQ , w] plane. In particular, assuming a scatter of 5% per redshift
bin in the angular diameter-redshift diagram (nearly corresponding to the scatter expected for a sample of 1300 rotators with
0 < z < 1.4, dz = 0.1, whose diameter is locally calibrated with
a 40% precision), we have shown that the input fiducial [ΩQ , w]
point is still within the 1σ error contours obtained by applying
the angular diameter test to the evolved data.
Finally, we have outlined the strategy to derive a cosmologyevolution diagram with which it is possible to establish an interesting mapping between different cosmological models and
the amount of galaxy disc/luminosity evolution expected at a
given redshift. The construction of this diagram does not require
an a-priori knowledge of the particular functional form of the
galaxy size/luminosity evolution. By reading this diagram, one
can infer cosmological information once a theoretical prior on
disc or luminosity evolution at a given redshift is assumed. In
particular if the amplitude of the relative disc evolution at z̄ = 1.5
is known to better than ∼30%, then an Einstein-de Sitter universe
(Ωm = 1) may be geometrically discriminated from a flat, vacuum dominated one (Ωm = 0.3, ΩQ = 0.7). Viceversa, one can
use the cosmology-evolution diagram to place constraints on the
amplitude of the galaxy disc/luminosity evolution, once a preferred cosmology is chosen.
In conclusion, given the simple ingredients entering the proposed implementation strategy, nothing, besides evolution of
discs, could in principle bias the test. Even so, evolution can
be easily diagnosed and, under some general conditions, it can
be shown that it does not compromise the possibility of detect-
C. Marinoni et al.: Geometrical Tests of Cosmological Models with High-z Galaxies. I.
ing the presence of dark energy and constraining the value of its
equation of state.
In the following papers of this series (Saintonge et al. 2007;
Marinoni et al. 2007), we implement the proposed strategy to a
preliminary sample of velocity-selected high redshift rotators.
Acknowledgements. We would like to acknowledge useful discussions with R.
Scaramella and G. Zamorani. This work has been partially supported by NSF
grants AST-0307661 and AST-0307396 and was done while A.S. was receiving
a fellowship from the Fonds de recherche sur la Nature et les Technologies du
Québec. K.L.M. is supported by the NSF grant AST-0406906.
Appendix A
Given a spectroscopically selected sample of objects with constant rotational velocity we can derive the observed magnitude
mo of a standard candle of absolute magnitude Mv (0) located at
redshift z, by using the standard relation Sandage (1972)
mo = mth (Mv (0), z, p) + M (z) + K(z)
(10)
where
mth = Mv (0) + 5 log dL (z, p) + 25
and were dL is the luminosity distance, K(z) is the K correction
term and M (z) is the a-priori unknown evolution in luminosity
of our standard candle, i.e. M (z) = ΔMv (z) = Mv (z) − Mv (0) is
the difference between the absolute magnitude of an object of
rotational velocity V measured at redshift z and the un-evolved
local standard value Mv (0).
From the definition of wavelength-specific surface brightness μ we deduce that the variation as a function of redshift in
the average intrinsic surface brightness (within a radius R) for
our set of homologous galaxies is
Δμth (z)R = ΔMv (< R) − 5 log
R(z)
·
R(0)
By opportunely choosing the half light radius Dv as a metric definition for the size of a galaxy we immediately obtain
(z, p)
θ
+1 ·
(11)
Δμth (z)D = M (z, p) − 5 log th
θ (D(0), z, p)
The intrinsic surface brightness evolution is not an observable,
but in a FRW metric this quantity is related to the surface brightness change observed in a waveband Δλ by the relation
Δμo (z)D = Δμth (z)D + 2.5 log(1 + z)4 + K(z).
(12)
Thus, once we measure the redshift evolution of Δμo (z)D for
the sample of rotators, the absolute evolution in luminosity corresponding to a given relative evolution in diameters can be directly inferred using Eq. (11).
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221
Astronomy & Astrophysics manuscript no. msIII
(DOI: will be inserted by hand later)
September 13, 2007
Geometrical Tests of Cosmological Models
III. The Cosmology-Evolution Diagram at z = 1
C. Marinoni1 , A. Saintonge2 , T. Contini3 , C.J. Walcher4 , R. Giovanelli2 , M.P. Haynes2 , K.L. Masters5 ,
O. Ilbert4 , A. Iovino6 , V. Le Brun4 , O. Le Fevre4 , A. Mazure4 , L. Tresse4 , J.-M. Virey1 , S. Bardelli7 , D.
Bottini8 , B. Garilli8 , D. Maccagni8 , J.P. Picat3 , R. Scaramella9 , M. Scodeggio8 , P. Taxil1 , G. Vettolani9 ,
A. Zanichelli9 , E. Zucca7
1
2
3
4
5
6
7
8
9
Centre de Physique Théorique? , CNRS-Université de Provence, Case 907, F-13288 Marseille, France.
Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
Laboratoire d’Astrophysique de l’Observatoire Midi-Pyrénées, UMR 5572, 31400 Toulouse, France
Laboratoire d’Astrophysique de Marseille, UMR 6110, CNRS Université de Provence, 13376 Marseille, France
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02143, USA
INAF-Osservatorio Astronomico di Brera - Via Brera 28, Milano, Italy
INAF-Osservatorio Astronomico di Bologna - Via Ranzani 1, Bologna, Italy
IASF-INAF - via Bassini 15, I-20133, Milano, Italy
IRA-INAF - Via Gobetti,101, I-40129, Bologna, Italy
Received ... / Accepted ...
Abstract. The rotational velocity of distant galaxies, when interpreted as a size (luminosity) indicator, may be
used as a tool to select high redshift standard rods (candles) and probe world models and galaxy evolution via the
classical angular diameter-redshift or Hubble diagram tests. We implement the proposed testing strategy using a
sample of 30 rotators spanning the redshift range 0.2 < z < 1 with high resolution spectra and images obtained by
the VIMOS/VLT Deep Redshift Survey (VVDS) and the Great Observatories Origins Deep Survey (GOODs). We
show that by applying at the same time the angular diameter-redshift and Hubble diagrams to the same sample
of objects (i.e. velocity selected galactic discs) one can derive a characteristic chart, the cosmology-evolution
diagram, mapping the relation between global cosmological parameters and local structural parameters of discs
such as size and luminosity. This chart allows to put constraints on cosmological parameters when general prior
information about discs evolution is available. In particular, by assuming that equally rotating large discs cannot
be less luminous at z = 1 than at present (M (z = 1) <
∼ M (0)), we find that a flat matter dominated cosmology
(Ωm = 1) is excluded at a confidence level of 2σ and an open cosmology with low mass density (Ωm ∼ 0.3) and
no dark energy contribution (ΩΛ ) is excluded at a confidence level greater than 1σ. Inversely, by assuming prior
knowledge about the cosmological model, the cosmology-evolution diagram can be used to gain useful insights
about the redshift evolution of baryonic discs hosted in dark matter halos of nearly equal masses. In particular, in
a ΛCDM cosmology, we find evidence for a bimodal evolution where the low-mass discs have undergone significant
surface brightness evolution over the last 8.5 Gyr, while more massive systems have not. We suggest that this
dichotomy can be explained by the epochs at which these two different populations last assembled.
Key words. cosmology: observations—cosmology:theory—cosmology:cosmological parameters—galaxies: distances
and redshifts—galaxies: fundamental parameters—galaxies: evolution
1. Introduction
Send offprint requests
to:
C.
Marinoni,
e-mail:
marinoni@cpt.univ-mrs.fr
?
Centre de Physique Théorique is UMR 6207 - “Unité Mixte
de Recherche” of CNRS and of the Universities “de Provence”,
“de la Méditerranée” and “du Sud Toulon-Var”- Laboratory
affiliated to FRUMAM (FR 2291).
Deep redshift surveys of the Universe, such as the
VIMOS/VLT deep redshift survey (VVDS, Le Fèvre et
al. (2005)) and the ACS/zCOSMOS survey (Lilly et al.,
2006) are currently underway to study the physical properties of high redshift galaxies. Motivated by these major
observational efforts, we are currently exploring whether
2
Marinoni et al.: The Cosmology-Evolution Diagram at z = 1
high redshift galaxies can also be used as cosmological
tracers. Specifically, we are trying to figure out if these
new and large sets of spectroscopic data can be meaningfully used to probe, in a geometric way, the value of
the constitutive parameters of the Friedmann-RobertsonWalker cosmological model.
A whole arsenal of classical geometrical methods has
been developed to measure global properties of the universe. The central feature of all these tests is the attempt
to probe the various operative definitions of relativistic
distances by means of scaling relationships in which a distant dependent observable, (e.g. an angle or a flux), is
expressed as a function of a distance independent fixed
quantity (e.g. metric size or absolute luminosity).
A common thread of weakness in all these approaches
to measure cosmological parameters using distant galaxies
or AGNs selected in deep redshift surveys is that there are
no clear criteria by which such cosmological objects should
be considered universal standard rods or standard candles.
Motivated by this, in previous papers (Marinoni et al.
(2004) and Marinoni et al. (2007), hereafter Paper I) we
have investigated the possibility of using the observationally measured and theoretically justified correlation between size/luminosity and disc rotation velocity as a viable
method to select a set of high redshift galaxies, with statistically homologous dimensions/luminosities. This set of
tracers may be used to test the evolution of the cosmological metric via the implementation of the standard angular
diameter-redshift and Hubble diagram tests.
Finding valid standard rods, however, does not solve
the whole problem; the implementation of the angular
diameter-redshift test using distant galaxies is hampered
by the difficulty of disentangling the effects of galaxy evolution from the signature of geometric expansion of the
universe.
In Paper I we have determined some general conditions
under which galaxy kinematics may be used to test the
evolution of the cosmological metric. We have shown that
in the particular case in which disc evolution is linear and
modest (<30% at z = 1.5), the inferred values of the dark
energy density parameter ΩQ and of the cosmic equation
of state parameter w are minimally biased (δΩQ = ±0.15
for any ΩQ in the range 0 < ΩQ < 1).
In Paper I, we also looked for cosmological predictions
that rely on less stringent assumptions, i.e. which do not
require specific knowledge about the particular functional
form of the standard rod/candle evolution. In particular,
we showed how velocity-selected rotators may be used to
construct a cosmology-evolution diagram for disc galaxies. This is a chart mapping the local physical parameter
space of rotators (absolute luminosity and disc linear size)
onto the space of global, cosmological parameters (Ωm ,
ΩQ ). Using this diagram it is possible to extract information about cosmological parameters once the amount
of size/luminosity evolution at some reference epoch is
known. Vice-versa, once a cosmological model is assumed,
the cosmology-evolution mapping may be used to directly
infer the specific time evolution in magnitude and size of
disc galaxies that are hosted in dark matter halos of similar mass.
We stress that this last way of reading the cosmologyevolution diagram offers a way to explore galaxy evolution which is orthogonal to more traditional methods. In
particular, insights into the mechanisms of galaxy evolution are traditionally accessible through the study of
disc galaxy scaling relations, such as the investigation
of the time-dependent change in the magnitude-velocity
(Tully-Fisher) relation (e.g., Vogt et al (1996); Böhm et
al. (2004); Bamford et al. (2006)), of the magnitude-size
relations (e.g., Lilly et al. (1998); Simard et al. (1999);
Bouwens & Silk (2002); Barden et al. (2005)), or of the
disc “thickness” (Reshetnikov et al., 2003; Elmegreen et
al., 2005). By applying the angular size-redshift test and
the Hubble diagram to velocity-selected rotators, we aim
at tracing the evolution in linear size, absolute magnitude
and intrinsic surface brightness of disc galaxies that are
hosted halos of the same given mass at every cosmic epoch
explored.
In this paper, we present a pilot observational program that allowed us to test whether galaxy rotational
velocity can be used to select standard rods, and to derive the cosmology-evolution diagram for disc galaxies at
redshift z = 1. Our observing strategy was to follow-up in
medium resolution spectroscopic mode with VIMOS a set
of emission-line objects selected from a sample of galaxies in the Chandra Deep Field South (CDFS) region for
which high resolution photometric parameters were available (Giavalisco et al., 2004).
The outline of the paper is as follow: in §2 we describe the VVDS spectroscopic data taken in the CDFS
region. In §3 we outline a strategy to test the consistency of the standard rod/candle selection. In §4 we derive the cosmology-evolution diagram for our sample of
rotators, and in §5 we present our results about disc size,
luminosity and surface brightness evolutions. Discussions
and conclusions are presented in §6 and §7, respectively.
Throughout, the Hubble constant is parameterized via
h70 = H0 /(70kms− 1M pc− 1). All magnitudes in this paper are in the AB system (Oke & Gunn, 1983), and from
now the AB suffix will be omitted.
2. Sample: observations and data reduction
Our strategy to obtain kinematic information for the
largest possible sample of rotators at high redshift was
to re-target in medium resolution mode (R=2500) galaxies in the CDFS region for which a previous pass in lowresolution mode (Le Fèvre et al., 2004) already provided
spectral information such as redshifts, emission-line types,
and equivalent widths, for galaxies down to I=24. Galaxies
were selected as rotators if their spectra was blue and characterized by emission line features (OII, Hβ, OIII, Hα).
CDFS photometry was then used to confirm the disc-like
nature of their light distribution (i.e. the absence of any
strong bulge component), and also to avoid including in
Marinoni et al.: The Cosmology-Evolution Diagram at z = 1
the sample objects with peculiar morphology or undergoing merging or interaction events.
The final sample of candidates for medium resolution
re-targeting was defined by further requiring that the inclination of the galaxy be greater than 60◦ to minimize
biases in velocity estimation, and that its identified emission line fall on the CCD under the tighter constraints imposed by the medium resolution grism. Once the telescope
pointing and slit positioning were optimized using the lowresolution spectral information, the remaining space on
the focal plane mask was blindly assigned to galaxies in
the field.
Spectroscopic observations have been obtained with
the VIMOS spectrograph on the VLT Melipal telescope
in October 2002. The slit width was 1 arcsecond giving a
spectral resolution R=2500 as measured on the FWHM
of arc lines. Using the VIMOS mask design software and
capabilities of the slit-cutting laser machine (Bottini et
al., 2005), slits have been placed on each galaxy at a position angle aligned with the major axis. The seeing at the
time of observations was 0.8 arcseconds FWHM with an
integration time of 1 hour split in three exposures of 20
minutes each.
Most of the galaxies in the CDFS area surveyed by the
VVDS have high resolution images taken with the ACS
camera of the HST by GOODs. Images are available in
four different filters (F435W, F606W, F775W, F850LP)
noted hereafter B,V,I and Z, respectively. A small fraction
of the targeted galaxies has only I band images provided
by the ESO Imaging Survey (Arnouts et al., 2001).
The galaxy rotational velocity has been estimated using the linewidths of the emission lines. A detailed analysis
of the velocity extraction algorithm and of the potential
systematic errors implicit in this technique are presented
in Paper II of this serie (Saintonge et al., 2007). This technique to measure rotation velocities imposed itself since
many galaxies at high redshift were too small to measure
rotation curves reliably, and since summing all the light
to form velocity histograms increased the signal-to-noise
ratio (S/N) of the detected lines.
Magnitudes have been computed in the I band and
a K-correction was applied (see Ilbert et al. (2005) for a
detailed discussion). They were also corrected for galactic absorption using the maps of Schlegel et al. (1998)
in the CDFS region (i.e. on average a correction of ∼
0.0016), and for galaxy inclination by adopting a standard empirical description of internal extinction Aλ in
the pass-band λ, γ log(sec i), where i is the galaxy inclination angle as calculated from the galaxy axis ratio and
γI = 0.92 + 1.63(log 2v − 2.5) (Tully et al., 1998) where v
is the maximum rotational velocity of a galaxy.
Galaxy sizes have been specified in terms of the halflight diameter (HLD) inferred in the I band. Typical errors in the measurements are σθ ∼ 0.04”. In Paper I we
stressed the importance of using a metric rather than an
isophotal definition of galaxy diameters for cosmological
purposes (e.g. Sandage, 1995). We also verified that the
HLDs for our sample of galaxies do not depend on wave-
3
length; there is no systematic difference in the inferred
metric diameters when the HLD is computed in the B,
V, I or z filters (see also Sandage & Perelmuter (1990), de
Jong (1996)). The scatter in the HLDs inferred in different
bands is of order 0.02” and therefore small in comparison
to the observational uncertainties σθ .
[OII] linewidths have been translated into an estimate of the galaxy rotational velocity, v, as detailed in
§3.2 of Paper II. Rotational velocity was derived using
[OII](3727Å) lines (24 objects), [OIII](5007Å) lines (10
objects) and Hα(6563Å) (5 objects). 23 galaxies have velocities in the range 0 < v( km s−1 ) ≤ 100 (with mean
velocity of the sample ∼ 60 km s−1 ) and 16 galaxies have
velocities in the range 100 < v(km s−1 ) ≤ 200 (with mean
velocity of the sample ∼ 143 km s−1 ) .
After data reduction, we were left with a sample of 39
objects, 27 of which have high resolution imaging. As for
the remaining objects with ground photometry, we only
consider in the following those with z < 0.2, in order to
exclude faint and small galaxies for which the size measurements are severely compromised by seeing distortions.
Therefore, our final ”science” sample contains 30 objects.
Data are organized and presented in Table 1 as follows:
col.1 : galaxy ID in the EIS catalog, col.2 : redshift, col.3
rotation velocity, col.4 : half-light angular radius, col.5 :
magnitude, col.6 : surface brightness within the half-light
radius.
3. Selection of Standard rods/candles
An observable relationship exists between the metric radial dimension D of a disc and its speed of rotation v. An
analogous empirical relationship connects rotation with
luminosity (Tully & Fisher, 1977). In Paper I we have
proposed to use information on the kinematics of galaxies,
as encoded in their OII emission-line width, to objectively
identify standard rods/candles at high redshifts. A discussion of the requirements and of the optimal strategies to
fulfill this observational program is detailed in Paper I.
A variety of standard rod candidates have been explored in previous attempts of providing a direct geometrical proof of the curvature of the universe. A common
thread of weakness in all these attempts is that there are
no clear physical nor statistical criteria by which the proposed objects (clusters, extended radio lobes or compact
radio jets associated with quasars and AGNs) should be
considered universal standard rods/candle.
Even assuming that a particular class of standards is
identified, the length of the rod remains unknown. Since
the inferred cosmological parameters heavily depend on
the assumed value for the object size (Lima & Alcaniz,
2000), an a-priori independent statistical study of the
standard rod absolute calibration is an imperative prerequisite. In Paper II, we used a large sample of galaxies
from the SFI++ catalog (Springob et al., 2007) to fix the
4
Marinoni et al.: The Cosmology-Evolution Diagram at z = 1
EIS ID
30445
32177
31328
32998
34826
34244
29895
37157
33200
33763
31501
31194
29342
29232
34325
34560
36484
16401
17811
17362
22685
17255
15152
15099
19702
16377
17421
20202
18416
17534
15486
19684
18743
15553
18417
18779
21252
20708
18853
z
0.9332
0.8934
0.4164
0.1464
0.4559
0.5321
0.6807
0.8677
0.1267
1.0220
1.0360
0.3320
0.4680
0.8610
0.3334
0.8618
0.7539
1.1000
0.8143
0.6814
0.8411
0.1787
0.7931
0.3661
0.6770
0.5621
0.7834
0.5763
0.8859
0.3493
0.6613
0.8588
0.6800
0.4584
0.5350
0.5623
0.5795
0.1228
0.6509
v(km s−1 )
97
68
79
28
204
55
44
129
96
130
140
80
55
155
26
28
25
306
170
178
115
99
169
70
62
36
99
26
99
35
146
104
81
183
36
59
102
166
116
θo (arcsec)
0.180
0.149
0.606
0.857
0.715
0.298
0.220
0.751
0.806
0.755
0.747
0.818
0.481
0.370
0.221
0.204
0.202
0.913
0.726
0.683
0.891
1.203
0.895
0.509
0.295
0.490
0.361
0.190
0.205
0.271
0.551
0.424
0.447
0.991
0.342
0.351
0.414
1.540
0.556
mo
23.744
23.681
22.297
20.794
22.684
22.324
23.571
23.399
20.519
23.404
23.513
21.950
23.565
22.320
23.570
23.549
23.781
21.628
23.573
22.379
22.147
21.758
21.702
21.171
22.812
22.944
23.280
23.318
23.637
23.837
22.495
22.790
22.950
19.292
22.877
22.445
22.854
18.427
21.350
µo (mag/arcsec2 )
22.03
21.54
23.20
22.45
23.95
21.69
22.23
24.77
22.04
24.79
24.87
23.51
23.97
22.15
22.20
22.00
22.30
23.42
24.87
23.54
23.88
24.16
23.46
21.70
22.16
23.37
23.06
21.70
22.18
22.99
23.20
22.92
22.70
21.26
22.54
22.15
22.93
21.40
22.10
Table 1. Properties of the Galaxy Sample
We have seen that, in order to implement the proposed
test, we need two sample of rotators: the “data sample”
(galaxies with the same rotational velocity selected over
the widest possible redshift range; the sample presented
in §2), and the “calibration sample” (rotators at redshift
z ∼ 0 for which the physical size of the linear diameter is
known; the SFI++ sample analyzed in Paper II). This last
sample allows us to calibrate the zero-point of the Hubble
and angular size-redshit diagrams (i.e. Mv (0) and Dv (0)
in eqs. 2 an 4).
indicators (spectroscopic lines) and two different velocity
extraction methods. Specifically we use OII linewidths to
measure the rotational velocity of the distant “data” sample and Hα rotation curves to measure the velocities of the
local “calibration” galaxies. Therefore, it is imperative to
check that possible biases or errors introduced by combining velocities inferred using systematically different measuring techniques do not prevent a meaningful comparison
between different samples at different redshifts.
To this purpose we have implemented the following
testing strategy. Given a spectroscopically-selected sample of standard candles Mv (0) with rotational velocity v,
one can derive the observed apparent magnitude mo of a
standard candle located at redshift z, by using the relation
(Sandage, 1988):
We stress that the disc rotational velocity of galaxies
in the two samples is measured using two different velocity
mo = mth (Mv (0), z, p) + M (z, p) + K(z)
local calibration values for absolute magnitudes and linear
diameters of galaxies with a given rotational velocity.
3.1. Velocity selection of rotators: test of consistency
(1)
Marinoni et al.: The Cosmology-Evolution Diagram at z = 1
where
m
th
= Mv (0) + 5 log dL (z, p) + 25
(2)
and where dL (z, p) is the luminosity distance (depending on the set of cosmological parameters p), K(z) is the
K-correction term and M (z, p) is the a-priori unknown
cosmology-dependent evolution in luminosity of our standard candle, i.e. M (z, p) = Mv (z, p) − Mv (0) is the difference between the absolute magnitude of an object of
rotational velocity v measured at redshift z with respect
to the un-evolved local standard value Mv (0).
Similarly, one can parameterize any possible evolution
affecting the standard rod Dv (0) by writing its observed
apparent subtended angle at redshift z as
θ0 = θth (Dv (0), z, p)[1 + δ(z, p)]
(3)
where the theoretically expected angular scaling (θ th ) is
given by
θth =
Dv (0)
,
dA (z, p)
dA = dL (1 + z)−2
(4)
and where δ(z, p) is a cosmology-dependent function
which describes the relative redshift evolution of the standard rod, i.e δ ≡ (Dv (z, p) − Dv (0))/Dv (0) ≡ D /Dv (0).
We note that any possible evolution in the standard rod
angular size is related to the evolution in its linear dimension as follows: θ = D /dA . Here and in the following, we
assume that the angular size of fixed-velocity rotators is
estimated using the galaxy half-light diameter Dv .
From the definition of wavelength-specific surface
brightness, µ, we deduce that the variation as a function of
redshift in the average intrinsic surface brightness within
a radius R for our set of velocity selected galaxies (i.e.
∆hµin (z)iR ≡ hµin (z) − µin (0)iR ) is
∆hµin (z)iR = ∆Mv (< R) + 5 log
R(z)
R(0)
(5)
By choosing the half-light diameter Dv as a metric
definition for the size of a standard rod, we immediately
obtain the intrinsic surface brightness evolution within Dv
as
∆hµin (z)iDv = M (z, p) + 5 log(1 + δ(z, p)).
(6)
While the specific amount of evolution in luminosity and
size do in principle depend on the specific background
cosmological model adopted, the corresponding evolution
in intrinsic surface brightness is a cosmology-independent
quantity.
The evolution in intrinsic surface brightness is not a
directly measurable quantity, but, in a FRW metric, this
quantity can be easily related to the apparent surface
brightness change observed in a waveband ∆λ by the relation
∆hµo (z)iDv = ∆hµin (z)iDv + 2.5 log(1 + z)4 + K(z)
(7)
We note that the left-hand side of Eq. 7 is directly measurable using photometric images. Moreover, it can be measured without assuming any specific galaxy light profile
5
and it will be, in general, a non linear function of redshift.
By combining eqs. 1, 3, 6 and 7 we define the η function:
η = mo (z) − ∆hµo (z)iDv + 5 log θo (z).
(8)
The specific combination in Eq. 8 of observed magnitudes, half-light diameters, and evolution in the observed surface brightness within the HLD (∆hµo (z)iDv =
hµo (z)iDv − hµo (0)iDv ) is, by construction, a redshiftinvariant quantity which is equal to
η = Mv (0) + 5 log Dv (0) + 25.
(9)
From a theoretical point of view, we emphasize that
the η-estimator given in Eq. 8 does not explicitly depend
on i) K correction, ii) evolution in luminosity or size of
our standard sources and iii) on the specific gravitational
model assumed to derive the exact functional form of the
angular and luminosity distances.
From an observational point of view, we stress that
Eq. 8 can be directly estimated using photometric images
of the “data” sample, while Eq. 9 may be expressed in
terms of the locally measured absolute magnitudes and
linear diameters of our “calibration” sample. Therefore,
by simply comparing the values of the η function inferred
using the “data” sample (Eq. 8) with the constant value
predicted using the “calibration” sample (Eq. 9), we can
test for the presence of eventual biases in our data. The
goal is to reveal possible systematics that could be introduced, for example, by the different techniques with which
rotation properties are inferred locally (mainly using Hα
rotation curves) and at higher redshift (mainly using OII
line-widths). Clearly, a mismatch would indicate that our
spectroscopic selection technique fails in selecting homologous classes of objects embedded in halos of nearly the
same mass at different redshifts.
Since our total sample is still limited, at present it is
practical to implement the proposed test of consistency by
defining only two broad classes of velocity-selected galaxies: a low-velocity sample of standard rods/candles with
0 < v ≤ 100km s−1 containing 22 galaxies with mean rotational velocity of ∼ 60 km s−1 (S60 sample) and a highvelocity set of objects with 100 < v < 200km s−1 containing 8 rotators with mean velocity of ∼ 143 km s−1 (S143
sample). The size (HLD), absolute luminosity and mean
surfac