Fluctuations in a quintessence universe [Elektronische Ressource] / presented by Khamphee Karwan

Dissertationsubmitted to theCombined Faculties for the Natural Sciences and for Mathematicsof the Ruperto-Carola University of Heidelberg, Germanyfor the degree ofDoctor of Natural Sciencespresented byMaster of Science-Physics: Khamphee Karwanborn in: Bangkok, ThailandOral examination: 14.02.2006Fluctuations in a Quintessence UniverseReferees: Prof. Dr. Christof WetterichProf. Dr. Matthias BartelmannFluktuationen in einem Quintessenz-UniversumZusammenfassungWir diskutieren die Entwicklung und E ekte von Quintessenz uktuationen ineinem FRW und in ation arem Universum. Nachdem wir die Prinzipien der kos-mologischen St orungstheorie eingefuhrt haben, geben wir Entwicklungsgleichungenfur metrische, Materie- und Quintessenz uktuationen an. Wir verwenden diese Gle-ichungen, um die Entwicklung von Quin in einem FRW Uni-versum zu studieren. Die Fluktuationen in einem Exponentialpotentialmodell mitnicht-kanonischem kinetischen Term k onnen das CMB Leistungsspektrum bei niedri-gen Multipolen sowohl erh ohen als auch erniedrigen, vorausgesetzt das Quintessen-zfeld bleibt bis heute eingefroren. In unserer Analyse ben otigen wir keinen Mechanis-mus zur Verst arkung der Feld uktuationen. Um zu ub erprufen, ob das Quintessen-zfeld bis heute eingefroren sein kann, betrachten wir dessen Entwicklung w ahrendder In ation. W ahrend der In ation wird der Erwartungswert des Quintessen-zfeldes zu gr osseren Werten hin verschoben.
Publié le : dimanche 1 janvier 2006
Lecture(s) : 36
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Source : ARCHIV.UB.UNI-HEIDELBERG.DE/VOLLTEXTSERVER/VOLLTEXTE/2006/6135/PDF/THESIS.PDF
Nombre de pages : 91
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Dissertation submitted to the Combined Faculties for the Natural Sciences and for Mathematics of the Ruperto-Carola University of Heidelberg, Germany for the degree of Doctor of Natural Sciences
presented by Master of Science-Physics: Khamphee Karwan born in: Bangkok, Thailand Oral examination: 14.02.2006
Fluctuations in a Quintessence Universe
Referees: Prof. Dr. Christof Wetterich Prof. Dr. Matthias Bartelmann
Fluktuationen in einem Quintessenz-Universum Zusammenfassung WirdiskutierendieEntwicklungundE ektevonQuintessenz uktuationenin einemFRWundin ationaremUniversum.NachdemwirdiePrinzipienderkos-mologischenStorungstheorieeingefuhrthaben,gebenwirEntwicklungsgleichungen furmetrische,Materie-undQuintessenz uktuationenan.WirverwendendieseGle-ichungen,umdieEntwicklungvonQuintessenz uktuationenineinemFRWUni-versum zu studieren. Die Fluktuationen in einem Exponentialpotentialmodell mit nicht-kanonischemkinetischenTermkonnendasCMBLeistungsspektrumbeiniedri-genMultipolensowohlerhohenalsaucherniedrigen,vorausgesetztdasQuintessen-zfeldbleibtbisheuteeingefroren.InunsererAnalysebenotigenwirkeinenMechanis-muszurVerstarkungderFeld uktuationen.Umzuuberprufen,obdasQuintessen-zfeldbisheuteeingefrorenseinkann,betrachtenwirdessenEntwicklungwahrend derIn ation.WahrendderIn ationwirdderErwartungswertdesQuintessen-zfeldeszugrosserenWertenhinverschoben.DadurchistderErwartungswertzu Beginn der Strahlungsdominierten Phase gross genug, um die Quintessenz bis heute eingefrorenzulassen.SchliesslichstudierenwirEinschrankungenandieEntwick-lung der Dunklen Energie durch Beobachtungsdaten. Wir verwenden hierzu eine Parametrisierung, welche von Wetterich vorgeschlagen wurde. Fluctuations in a Quintessence Universe
Abstract Inthisthesis,wediscusstheevolutionande ectsofquintessence uctuationsin aFRWandin ationaryuniverse.Afterintroducingthefundamentalideasofcos-mological perturbation theory, we give the evolution equations for metric, matter andquintessence uctuations.Weusetheseequationstostudytheevolutionand e ectsofquintessence uctuationsinaFRWuniverse.The uctuationsinanexpo-nential quintessence model with non-canonical kinetic term can suppress or enhance theCMBpowerspectrumatlowmultipoles,ifthequintessence eldisfrozenuntil the present epoch. In our analysis, we do not need any mechanism for amplifying the eld uctuations.Tocheckwhetherthequintessence eldcanbefrozenuntil thepresentepoch,weconsideritsevolutionduringin ation.Duringin ation,the meanvalueofthequintessence eldisdriventowardsalargevaluebyitsquan-tum uctuations.Asaresult,thevalueofthequintessence eldatthebeginning ofradiationdominationislargeenoughtokeepthequintessence eldfrozenuntil the present epoch. Finally, we study observational constraints on the dark energy evolution using a parameterization proposed by Wetterich.
Contents
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Introduction 1.1 FRW Universe . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Standard Cosmological Model . . . . . . . . . . . . . . . . 1.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 About this Thesis . . . . . . . . . . . . . . . . . . . . . . .
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Cosmological Perturbation Theory 2.1 Gauge Invariant Perturbation Variables . . . . . . . . . . . . 2.1.1 Metric Decomposition . . . . . . . . . . . . . . . . . 2.1.2 Energy Momentum Tensor . . . . . . . . . . . . . . . 2.1.3 Harmonic Decomposition . . . . . . . . . . . . . . . . 2.1.4 Gauge Transformations . . . . . . . . . . . . . . . . . 2.1.5 Gauge Invariant Variables . . . . . . . . . . . . . . . 2.2 Evolution Equations for Scalar Perturbations . . . . . . . . . 2.2.1 Conservation of the Energy Momentum Tensor . . . . 2.2.2 Einstein Equations . . . . . . . . . . . . . . . . . . . 2.2.3 Klein-Gordon Equations . . . . . . . . . . . . . . . . 2.3 Entropy and Adiabatic Perturbations . . . . . . . . . . . . . 2.3.1 Entropy Perturbations . . . . . . . . . . . . . . . . . 2.3.2 Adiabatic and Isocurvature Conditions . . . . . . . . 2.3.3 Correlated Adiabatic and Isocurvature Perturbations 2.4Evolution Equations for Matter and Radiation. . . . . . . . . . 2.4.1 Photons and Massless Neutrinos . . . . . . . . . . . . 2.4.2 Cold Dark Matter and Baryons . . . . . . . . . . . . 2.4.3 Dark Energy . . . . . . . . . . . . . . . . . . . . . . .
Fluctuations in Quintessence 3.1 Evolution of Tracking Quintessence . . 3.2 Superhorizon Scales Fluctuations . . . 3.3 Subhorizon Scales Fluctuations . . . . 3.4 E ects of Quintessence Fluctuations on
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In ation . . . . . . . . . . . . . . .
Initial Conditions from 4.1 Stochastic Approach 4.2 Classical Evolution . 4.3 Quantum Evolution .
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Observational Constraints on Dark 5.1 Dark Energy Parameterization . . 5.2 SNe Ia constraints . . . . . . . . 5.3 CMB plus LSS constraints . . . .
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CONTENTS
Chapter
1
Introduction
Before the 17th century, attempts to understand the universe were based on philo-sophicalpointofview.Thescienti cperceptionsoftheuniversehavestartedafter Newton proposed the law of gravity. Because of the limits of observational informa-tions, those perceptions were not quite right. At the beginning of the 20th century, Einstein used the theory of general relativity to construct the model of the universe. Since people believed that the universe is static, Einstein introduced the cosmolog-ical constant to balance the gravitational attractive force due to the matter in the universe. In 1929, Hubble measured distance-redshift relation of galaxies and found that the redshift of light emitted from galaxies increases with their distance so the universe is expanding. Thus, the cosmological constant was not necessary because the Einstein equations can give rise to the expanding universe. Many theories for the expandinguniversewereproposedbutsomeofthemhavebeenfalsi edbycurrent observations. Recently the instruments and techniques for observing the universe have been improved and the picture of our universe is more clear than 20 years ago. Cosmology now is in the stage of “Modern Cosmology”. In this chapter, we will give a brief overview of cosmology. Observations currently suggest that the expansion of the universe is accelerat-ing at the present epoch [1]. Since the cosmological constant can give rise to the accelerating universe, it plays a crucial role in modern cosmology. However, the origin of the cosmological constant is mysterious because its magnitude is extremely small compared with the energy scale at the time when it should originate. This is the cosmological constant problem [2, 3]. Because of the cosmological constant problem, a mysterious form of energy, called dark energy, has been suggested [4]-[12] forexplainingtheacceleratedexpansionoftheuniverse.Theevolvingscalar eld, i.e., quintessence, is a possible candidate for dark energy. The cosmological model is called Lambda Cold Dark Matter Model (CDM model) if the cosmological constant drives the accelerated expansion of the universe, and called Quintessence Cold Dark Matter Model (QCDM model) if quintessence drives the accelerated expansion. The
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CHAPTER 1.
INTRODUCTION
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recent observational data cannot be used to distinguish the cosmological constant from the evolving dark energy [13]-[15]. Moreover, many high energy physics mod-els of dark energy cannot be ruled out by the current observations. At the present epoch, dark energy constitutes about 60% 70% of the total energy in the universe [16] while the remainder is mainly contributed by dark matter. Dark matter is also a mysterious object in the universe. It has been introduced for explaining the ro-tation of galaxies [17]. In the simplest case, we expect that the circular velocityv of matter, which orbits around the center of a galaxy, should follow Kepler’s law vr 1/2, wherer The surprising resultis the distance from the center of galaxy. from measurements of galaxy rotation curve is that the velocity does not follow the r 1/2 This implies that the mass of galaxies are largerlaw, but stays constant. than we can observe. This missing mass is non-luminous. Thus, it is called dark matter. There are many candidates for dark matter, for example, axions, axinos, massive neutrinos, etc, but the massive neutrinos candidate has been ruled out by observations [18]. In section 1.1, we consider the dynamics of the Friedmann-Robertson-Walker (FRW) universe. We give a brief overview of the standard cosmological model in section 1.2. We summarize some observational constraints on cosmological models in section 1.3.
1.1 FRW Universe
In the theory of general relativity, gravity is viewed as a manifestation of spacetime curvature. The action of gravity on matter is described by the Einstein equation [19] 1
R= 8 G G=R 2gT,(1.1) whereGnsor,gric tensor andGis Newton’s constant. is the Einstein te is the met The Ricci scalarRand Ricci tensorRcorrespond to the curvature of spacetime,  while the energy momentum tensorTdescribes the energy and momentum of the matter in the spacetime. According to the observations, the universe looks rather homogeneousandisotropiconlargescales.Onemightthinkat rstsightthat the homogeneous universe should be isotropic, but it is not really true when we apply these notions to the spacetime of the universe. Because the universe evolves in time, the universe is homogeneous and isotropic in space, but not in time. A space manifold, such asRS2, can be homogeneous but nowhere isotropic, while a cone is isotropic around its vertex but not necessarily homogeneous. If the universe is isotropic around one point and also homogeneous, it will be isotropic around every point. This means that there is no special point in the universe, i.e. the universe looks the same around every point. Hence, we assume both homogeneity
CHAPTER 1.
INTRODUCTION
and isotropy. This is the cosmological principle.
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The homogeneity and isotropy of
the space imply that the space must be maximally symmetric [19]. Using this fact, one can derive the metric tensor for the spacetime of the universe. The line element can be written as 2 ds2= dt2+a2(t)1 drKr2+r2(d2+ sin2 d2),(1.2)
whereais the scale factor, andr, and Thisare the comoving coordinates. is the Friedmann-Robertson-Walker metric parameter. TheKdenotes the curvature on spatial hypersurfaces. The casesK<0,K= 0, andK>0 correspond to constant negative, no, and positive curvature on spatial hypersurfaces, respectively. Usually, these cases are called open forK<0, at forK= 0, and close forK>0. For the at case, the metric is ds2= dt2+a2dr2+r2d2+r2sin2 d2= dt2+a2dx2+dy2+dz2.(1.3)
The spatial part is simply at Euclidean space. We will see that the at case is suggested by observations. The Ricci scalar and Ricci tensor can be computed using the metric given by eq. (1.2).
We next consider the possible forms of energy and matter in the universe. The energy and matter in the universe can be treated as a perfect uid which is de ned as a uid that is isotropic in its rest frame [19, 20]. Since a perfect uid is at rest in comoving coordinates, its 4 velocity isu= (1,0,0,0). Using the the de nition of energy momentum tensor in eq. (2.7), we obtain T=0gi0jp,(1.4)
hereis the energy density andp this equationis the pressure of the uid. In we have neglected the anisotropic stress tensor because the background universe is isotropic. From the zero component of the conservation of energy equation, i.e., rT0= 0, we obtain 0= 3H(1 +w),(1.5)
where a prime denotes the derivative with respect to conformal time,w=p/is the equation of state parameter andH=a0/a conformal time. The asis de ned ad=dt the case when Inuse these notations throughout this thesis. will . Wewis constant, the above equation gives
a 3(1+w) .
(1.6)
The simplest perfect uids are matter and radiation. Matter is collisionless, nonrel-ativistic particles, whosew Radiation= 0. is the relativistic particles,e.g., photons,
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