Quintessence and Quantum Corrections

Transcription

1 Quintessence and Quantum Corrections DIPLOMARBEIT von Mathias Garny 9. Dezember 2004 Technische Universität München Physik-Department T30d Prof. Dr. Manfred Lindner

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3 Contents Introduction 2 Cosmology and Quintessence 5 2. Motivations for Dynamical Dark Energy The Framework of Cosmology Cosmology with a Scalar-Field Useful forms of the Equation of Motion Constraints on Dynamical Dark Energy Dark Energy in the Early Universe Equation of State Inhomogeneities Quintessence Dynamics 5 3. Models with Exponential Potential Single Exponential Multiple Exponentials Models with Inverse Power Law Potential Attractors of Inverse Power Laws Dark Energy Model with Inverse Power Laws Tracking Solutions The Tracker Condition Dynamic Mass Scale The Prototype Potentials Alternative Potentials Extended Quintessence Yukawa Coupling Varying Fundamental Constants Non-Minimal Couplings Possible Origins of the Quintessence Action Fermion Condensate Conformal Anomaly Renormalization of the Quintessence Potential The Effective Potential Derivation of the Effective Potential Effective Action and N-Point Functions The One-Loop Effective Potential Regularization of the Effective Potential Remarks on the Renormalizability iii

4 iv CONTENTS 4.2 Renormalization Group Equations Feynman Rules Renormalization to One-Loop Order Derivation of the Renormalization Group Equations Renormalization Group Improved Effective Potential Callan-Symanzik Equation for the Effective Potential The Leading Logarithm Potential RGE for the Leading Logarithm Potential Illustration in the Case of φ 4 Theory Alternative Approaches to the RGEs RGEs and the one-loop Effective Potential Translational Invariance of the RGEs Stability of Quintessence The Quintessence Scenario Orders of Magnitude for Quintessence Power-Counting Rules Space-time Derivatives in the Effective Action Heat Kernel Expansion Result for the Effective Action Effective Potential Component Logarithmic Component Higher Components Quadratic Divergences Quartic versus Quadratic Divergences Special Potentials Running of the Potential Running of the Effective Action Exponential Potential Inverse Power Law Potential Asymptotical Flatness and Conformal Anomaly Curved Space-Time Scenario Effective Action in Curved Space-Time One-Loop Effective Action RGEs in Curved Space-Time Conclusions 99 A Conventions and Constants 0 B Dimensional Regularization 03 C Heat Kernel Expansion 07 Bibliography 3 Danksagung 9

5 Chapter Introduction The universe has fascinated philosophers and scientists since thousands of years. Even though the horizon has been enhanced with time, the question about the fundamental components from which our cosmos is made of is unanswered until today. Aristoteles tried to explain our world by the elementary contents fire, water, soil and air. However, the cosmos is not complete unless the fifth, and, following Aristoteles, most important element is added: the Quintessence, which is omnipresent in the universe. This aether has been interpreted by physicists as a medium carrying electromagnetic waves, a theory which has been discarded over one hundred years ago. However, the original over 2300 year old basic idea of Quintessence could come to a renaissance today. Modern astro particle physics still cannot answer the question about the fundamental components from which our universe is built. However, especially in the last decade some enormous progress has been made. One can thereby view it as the greatest insight that we know what we do not know yet. Many astronomical observations of the luminosity of distant Supernovae a, fluctuations in the cosmic microwave background and the large scale structure of the galaxy distribution give increasing evidence [53,55,62] that pressure-less matter makes up only less than one third of the total energy density in the universe. It is one of the most fascinating challenges to explore the essence of the unknown form of energy which makes up the missing and largest part, over two thirds, of the cosmos: the so-called dark energy. The overall expansion of the universe, which has been revealed in 930 by Edwin Hubble, is being described by Einstein s general theory of relativity. Originally, Albert Einstein introduced a cosmological constant in his equations to obtain static solutions. After Hubble s discovery Einstein discarded the cosmological constant as the greatest blunder of my life. Today measurements of the redshift of distant SNa show that the expansion of the universe is even accelerated [62]. The gravitational effect of pressureless matter cannot cause such a behavior, which leads to the introduction of a new form of energy. The only source of knowledge about this dark energy is its gravitational interaction, which is supposed to be fundamentally different from usual matter: Today, dark energy is a repulsive source of gravity. This repulsion should power the accelerated expansion of our universe. Furthermore, it is observed that only one third of the energy density in the universe is concentrated in clustered structures like galaxies, stars and planets. Therefore the rest, the dark energy, has to be distributed homogeneously over the universe [78]. The simplest explanation of dark energy consists in a re-introduction of the cos-

6 2 INTRODUCTION mological constant, which acts as a counter-force to gravity. Indeed there are several sources for such a constant: Quantum field theory predicts that even the vacuum has nontrivial physical properties. The energy density of the vacuum could contribute to the dark energy in form of a cosmological constant. However, the vacuum energy is dominated by the most energetic fluctuations of the quantum fields. Cutting off the vacuum fluctuations at the Planck scale where quantum field theory is supposed to loose its validity leads to a dark energy density which is 23 orders of magnitude higher than the measured value. Furthermore, at each phase transition of the early universe contributions to the cosmological constant are expected which are far too large. Terms of this order of magnitude cannot contribute to the expansion of the universe. Otherwise, the universe would expand so fast that no galaxies and stars could have formed. How can dark energy be explained then? One possibility is to consider a cosmological constant of exactly the order of magnitude which is required for dark energy today. This would mean that there has been an enormous hierarchy between the dark energy content and the relativistic matter content in the very early universe. Apart from the fact that this requires enormous fine-tuning it appears in this case absolutely arbitrary that the dark energy density and the matter density are of the same order of magnitude exactly today. Therefore trying to explain the dominant form of energy in our cosmos more satisfactorily by considering alternative dark energy models seems to be well justified. One attempt is to find a natural explanation for the order of magnitude of a cosmological constant (e.g. by models involving extra-dimensions), another is to explain the accelerated expansion by a modification of the basic equations of general relativity. Furthermore, there is a whole class of models that explains dark energy, like the other elementary components of the universe, by a dynamical degree of freedom (see [53,55,65,67] for general reviews on all issues). The smallness of the dark energy in comparison to the Planck scale is then simply explained by the huge age of the universe. Of course it is not only expected from such models that they explain all experimental data but also allow viable solutions with a more natural choice of parameters than for a cosmological constant. Quintessence is a promising model in this context. The accelerated expansion of the universe is being described by a cosmological scalar-field which, in analogy to common models of the inflationary phase in the early universe, slowly rolls down a potential. A broad class of Quintessence models possesses so-called tracking solutions with some very nice properties: Due to the attractivity of the tracking solutions in phase space the present evolution of the universe is independent from the dark energy content in the early universe, which therefore does not have to be fine-tuned to an extraordinarily small number. Furthermore, the comparable orders of magnitude of dark energy and matter can be explained naturally since both components evolve in a similar way during cosmic expansion. The accelerated expansion is explained by the fact that the tracking attractor gets unstable today, triggered either naturally by the dominance of dark energy or by a change in the slope of the potential (depending on the specific model), and the system moves towards a new attractor where dark energy decays only very slowly and the universe accelerates. These Quintessence models will be presented in detail in the first part of this work and their properties will be disussed extensively, including some extensions like a direct coupling of Quintessence with matter and nonminimal coupling with gravity. Some of the most representative Quintessence models with tracking solutions involve e.g. exponential and inverse power law potentials. Such potentials are very unusual

7 CHAPTER. INTRODUCTION 3 from the point of view of high energy physics. For increasing field values, they decay over many orders of magnitude and asymptotically get flat and approach zero. If such potentials really exist in nature, they should be relics from a superior theory (possibly involving quantum gravity) in the very early universe, which makes Quintessence models interesting also for high energy physics. Thus the Quintessence field could be an effective field from integrating out some high energy degrees of freedom of an unknown theory at scales up to the Planck scale. Although the underlying theory is unknown, the low energy regime (i.e. at energy scales much below the Planck scale) should be described by quantum field theory. The very unusual and generally non-renormalizable form of the potentials used in Quintessence theories therefore rises the question, whether these models are stable under quantum corrections. The effects of the quantum corrections on the classical dynamics, which are described by the effective action, can in principle be quite severe: The effective action does not need to have the same symmetries as the classical pendant and can involve structurally new terms that may alter the properties and the existence of certain solutions. For example, the conformal anomaly could have impact on the asymptotical flatness of the potential. Therefore an investigation of quantum corrections to Quintessence models, which is the aim of the present work, seems to be very well motivated. Of course, the nonrenormalizability in the ususal power-counting sense sets tight limits to such an attempt. It is not possible to discuss quantum corrections to the same precision as in usual high energy physics, but only on a much more basic level. However, some aspects can be translated to this case and used for a study of Quintessence models. The aim is to constrain the functional form of possible quantum corrections by a very general analysis of subsets of operators that close under quantum corrections and to investigate the effects of the conformal anomaly by a renormalization group treatment. Quantum corrections to the Quintessence potential as well as those to the kinetic term will be investigated. The analysis will also be performed in a curved space-time setting using the semi-classical approximation for gravity and giving the possibility to investigate nontrivial couplings between the Quintessence field and geometrical quantities. Organization of this work The next chapter is devoted to the fundamental cosmological questions, equations and constraints. Chapter 3 introduces some common Quintessence models, general requirements for tracking behavior and treats extensions and possible origins of Quintessence. In Chapter 4 the renormalization group equations for the Quintessence potential are obtained by own calculations based on standard techniques and with special emphasis on the fact that the final results are independent of the Taylor expansion which is used during calculations. Chapter 5 investigates the relevance of the previous results for Quintessence models as well as corrections to the kinetic term and from curved spacetime. The conventions and notations which are used are listed in appendix A, while the other appendices contain further techniques and calculations as referenced in the text.

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9 Chapter 2 Cosmology and Quintessence In the first section of this chapter, the striking cosmological questions related to dark energy will be formulated precisely, and it will be discussed in how far dynamic dark energy models can address these questions. After that, the basic framework describing cosmology with a Quintessence field will be introduced. In the last section, observational constraints will be specified that can restrict the properties of dark energy. 2. Motivations for Dynamical Dark Energy Since there is strong observational evidence that the expansion of the universe is accelerated today [33, 62], the question about a cosmological constant has become urgent again. A cosmological constant is actually expected from the vacuum mode summation in quantum field theory as well as from potential shifts, e.g. during electroweak symmetry breaking. However, these contributions are far too large not only from the point of view of direct observations, but would also lead to a cosmos which is structurally quite different from the clustered universe in which we live. Furthermore, the value of the energy density of a cosmological constant as required by observations happens to be at exactly the same order of magnitude as the contribution from cold dark matter. In order to be able to distinguish clearly between the different cosmological questions it is necessary to make a detailed definition: QFT smallness problem: Why is there no huge cosmological constant contributing an vacuum energy density of order M 4 pl, M 4 GUT, M 4 SUSY or M 4 el. weak? Cosmological smallness problem: How can one explain a small nonzero cosmological constant or dark energy density? Coincidence of scales: The present dark energy and matter densities are ϱ DE M 4 pl and ϱ M M 4 pl. (2.) Coincidence of epochs: In our present cosmological epoch the expansion of the universe changes from decelerated to accelerated [62]. The last two items are observational statements. The question is whether there is a natural explanation for these coincidences or whether they are just an accident. The values are to be understood as a rough illustration of the orders of magnitude. They are based on the concordance model Ω DE = 0.7, Ω M = 0.3 and use H 0 = 70 km/s Mpc.

10 THE FRAMEWORK OF COSMOLOGY It appears likely that these questions cannot be answered by a single approach. On the one hand, a mechanism (or a symmetry) is needed that explains why the huge field theoretical contributions including contributions from potential shifts do not exist at all or at least why they do not act as a source of gravity. On the other hand, the observed acceleration of the universe has to be explained. The ΛCDM scenario with a cosmological constant is in accordance with practically all present observations inside the errorbars [68]. However, it does not answer any of the four cosmological questions above. The value of the cosmological constant has to be fine-tuned to fulfill the two coincidences : At the Planck epoch there is a hierarchy of order 0 23 between the energy density of the cosmological constant and the relativistic matter content in this model. Starting point for dynamical dark energy models is the Cosmological smallness problem. The aim is to explain the smallness of dark energy by the huge age of the universe. Therefore a time-dependent cosmological constant can be introduced that decays (similar to matter or radiation density) during cosmic evolution thus providing a natural explanation for its smallness today. At the Planck scale the dark energy content of the universe does not have to be fine-tuned to an extraordinarily small number. General covariance of the equations of motion dictates that the dark energy cannot only depend on time but is given by a space-time dependent field 2 which has to be added to the Lagrangian of the theory as a new dynamical degree of freedom. This opens up a whole field of possibilities manifesting themselves in a huge variety of scalar-field-based models, like Chaplygin Gas (a cosmic fluid derived from a Born-Infeld Lagrangian with a strange equation of state p /ϱ), Phantom energy (derived from a scalarfield Lagrangian with a kinetic term with a wrong sign and with a pressure p < ϱ) or k-essence (with a nonlinear kinetic term) and of course, most straightforward and probably most elaborated, Quintessence with a standard kinetic term and a selfinteraction described by the Quintessence potential, to name only a few (see [65] for a recent review, [4]). The details of the decaying field are important when addressing the coincidence of scales. Generally, it will therefore depend on the specific model in how far a natural explanation for this remarkable coincidence is found. Quintessence provides a special class of so-called tracking solutions that accounts for this coincidence, which will be discussed in the next chapter. The coincidence of epochs is not generically addressed by dynamic dark energy models. In some models the two coincidences are linked (like for a cosmological constant), while in other models they have to be discussed separately. 2.2 The Framework of Cosmology The framework of cosmology is the general theory of relativity, and cosmological models with dynamical dark energy can be formulated within this setting. However, one should keep in mind that cosmology is based on some fundamental assumptions, like isotropy and large-scale homogeneity of all components of our universe. Their validity is assumed in the following. Scalar-fields in cosmology are actually not unusual. Already shortly after the big 2 Just replacing the cosmological constant by a function Λ(t) is not possible because the Einstein equations can only be solved for covariant conserved energy-momentum tensors T µν;ρ = 0. However, (Λg µν) ;ρ = 0 only if Λ const.

11 CHAPTER 2. COSMOLOGY AND QUINTESSENCE 7 bang the universe may have undergone an accelerated phase, the cosmic inflation, which is often described by a slowly rolling scalar-field, called inflaton (see e.g. [49]). In this section, the Quintessence scalar-field will be introduced into the general theory of relativity, in close analogy to the inflaton scalar-field 3. The basic equations for the evolution of the universe will be presented starting from the Einstein action together with the Quintessence action. The equation of motion for the Quintessence field φ will be developed and some important parameters, like the equation of state parameter ω, will be defined. This is the basis for the investigation of Quintessence models Cosmology with a Scalar-Field The starting point is the gravitational action with a standard kinetic term and a potential for the Quintessence scalar-field φ given by [60, 75, 80] S = d 4 x g(x) ( 6 M 2pl R + 2 ) ( φ)2 V (φ) + L B, (2.2) where M pl is the reduced Planck mass defined as M pl 2 /6 /6πG, G is Newton s constant and L B is the Lagrangian describing all other forms of energy like dark matter, baryonic matter, radiation and neutrinos, which will be called background. Furthermore, g(x) is the determinant of the metric g µν (x) and R is the curvature scalar as defined in (A.4). The coupling of the Quintessence field to gravity is called minimal in this case since there are no explicit coupling terms like φ 2 R. It is only mediated through the determinant g(x) and the scalar product ( φ) 2 = µ φ ν φ g µν in the kinetic term. Possible constant contributions in the action (i.e. the cosmological constant) are assumed to be absorbed into the potential V (φ). Variation of the action with respect to g µν yields the Einstein equations R µν R 2 g µν = 8πG(T B µν + T Q µν ), (2.3) with the Ricci-tensor R µν (defined in (A.3)), the energy-momentum tensor for the background Tµν B = 2 δ( gl B ) g δg and µν ( ) Tµν Q = µ φ ν φ g µν 2 ( φ)2 V (ϱ φ + p φ )u µ u ν g µν p φ. (2.4) The energy-momentum tensor can be expressed in analogy to a perfect fluid with unit 4-velocity vector u µ = µ φ/ ( φ) 2 and energy density and pressure given by ϱ φ = 2 ( φ)2 + V (φ) and p φ = 2 ( φ)2 V(φ). (2.5) Variation of the action with respect to φ leads to the equation of motion for the 3 One could ask whether the inflaton and the Quintessence field could be identical. It is indeed possible to construct such models [56, 63]. The basic problem is that on the one hand the inflaton has to be coupled to matter strongly enough to reheat the universe after inflation, while on the other hand Quintessence may only be coupled very weakly (see section 3.4.). Anyway, the discussion of the Quintessence regime is basically not influenced by the role of the scalar field during inflation.

12 THE FRAMEWORK OF COSMOLOGY Quintessence field 4 φ + with the covariant d Alembertian for a scalar-field dv (φ) dφ = 0 (2.6) = D µ D µ = g µ g µ. (2.7) Under the assumptions of isotropy, homogeneity and a spatially flat universe 5 the Robertson Walker Metric (with k = 0) for comoving coordinates x µ = (t, x) with a dimensionless scalefactor a(t) can be used: ds 2 = g µν dx µ dx ν = dt 2 a(t) 2 d x 2. (2.8) After specializing the energy-momentum tensors to contain only space-independent densities ϱ B (t) and ϱ φ (t) and pressures 6 p B (t) and p φ (t) the Einstein equations reduce to the Friedman equations M pl 2 H2 = ϱ φ + ϱ B (2.9a) M pl 2 ä a = 2 (ϱ φ + 3p φ + ϱ B + 3p B ) (2.9b) with the Hubble parameter H = ȧ/a. The critical density is defined as ϱ c M 2 pl H2 = 3H 2 /8πG. The first Friedman equation is often written in terms of Ω i ϱ i /ϱ c = Ω φ + Ω B. (2.0) In the case of a homogeneous field φ(t) the covariant d Alembertian is (note that g = a 6 ) = a 3 t a 3 t = 2 t + 3H t, (2.) yielding an equation of motion from (2.6) for the homogeneous Quintessence field: φ + 3H φ + dv (φ) dφ = 0. (2.2) Speaking illustratively, one can imagine that a force dv/dφ accelerates the φ-field towards smaller potential energies thereby being damped by the 3H φ-term. However, the damping depends on the contents of the universe including Quintessence itself which means there is a back-reaction. The equation of motion is equivalent to the first law of thermodynamics d(a 3 ϱ φ )/dt = p φ da 3 /dt, (2.3) 4 If the background Lagrangian L B contains φ (e.g. Quintessence-dependent couplings) the right hand side of the equation of motion has to be replaced by δl B/δφ. For the basic discussion of Quintessence it will be assumed that this term has a negligible influence on the dynamics of the φ field. 5 An open or closed universe would not lead to different equations of motion for the φ field. Therefore the restriction to a flat universe as predicted from inflation and measured within some percent from the CMB is done already here, since all calculations will be made just for the flat case anyway. 6 The energy momentum tensors for the background and the φ field are assumed to be of the form of an ideal fluid Tµν i = (ϱ i + p i)u µu ν g µνp i with u µ = (, 0).

13 CHAPTER 2. COSMOLOGY AND QUINTESSENCE 9 Figure 2.: Schematic illustration of the equation of motion of the Quintessence field. [7] which can also be obtained from the requirement of covariant conservation of the energymomentum tensor Tµν Q ;ν = 0. Actually, this law is also valid for each independent 7 species i in the background d(a 3 ϱ i )/dt = p i da 3 /dt. (2.4) Furthermore it can be shown that the corresponding equation for the total energy density ϱ total ϱ φ + i ϱ i and the (analogically defined) total pressure p total can be derived from the Friedman equations. Thus, assuming N species in the background, there are 4 + N independent equations (second order differential equations are counted twice) from (2.4, 2.2, 2.9) with 4+2N independent variables a, ȧ, φ, φ, ϱ i, p i. This means the system can only be solved by specifying N additional equations, conventionally taken to be the equations of state for the N background species: p i = p i (ϱ i ) ω i ϱ i. (2.5) A constant equation of state parameter ω i, together with the first law of thermodynamics (2.4), yields the scaling behavior of the most important background components 8 ω M = 0 ϱ M a 3 nonrelativistic matter, ω R = /3 ϱ M a 4 relativistic matter, (2.6) ω Λ = ϱ Λ a 0 cosmological constant. It is useful to define the equation of state parameter ω φ in analogy to the background for the Quintessence field: ω φ = p φ = φ 2 /2 V ϱ φ φ 2 /2 + V. (2.7) However, the crucial difference is that this parameter will in general not be a constant. Therefore the scaling behavior of Quintessence cannot be integrated as easily as for matter and radiation. Like in inflationary scenarios, it is used that ω φ can be close to if the scalar-field is slowly rolling (i.e. φ2 /2 V ) down its potential. It can 7 An independent species should have negligible interaction with other species. 8 The cosmological constant is only given for completeness. It does not appear in the background since it is absorbed into the potential V.

14 THE FRAMEWORK OF COSMOLOGY be seen from the second Friedman equation (2.9b) that it is a necessary condition for an accelerated expansion of the universe that ω φ < /3. If the Quintessence field is static ( φ = 0), it acts like a cosmological constant V with ω φ =. On the other hand, a freely rolling field ( φ 2 /2 V ) has ω φ = + and scales like a 6. In any intermediate case one has ω φ + (2.8) if the potential is positive. Models with ω φ < can be obtained by flipping the sign of the kinetic term in the Lagrangian (Tachyonic or Phantom dark energy) or by introducing new terms in the action, leading to cosmologies with a Big Rip in the future. Such models allow superluminal velocities (see section 2.3.2) and are unstable on the quantum level since the energy density is not bounded from below [65]. We will not consider these models Useful forms of the Equation of Motion With the information given above it is possible to solve the equations of motion if the potential V (φ) and initial conditions for a, ȧ, φ, φ and ϱi are specified. However, it turns out that it is easier to switch from the cosmic time t to the scalefactor a as an independent variable. The 4 + N independent differential equations then split up into two first order differential equations with τ = ln dx dτ dy dτ and the + N algebraic equations = 3x h du dy (2.9a) = h x (2.9b) a a ref, y = φ/ M pl, x = φ/ M pl H ref, U = V/ M 2 pl H2 ref (2.20) h(x, y, a) = H H ref = H ref Mpl 2 x2 + U(y) + κ B (a), (2.2) ϱ i (a) = ϱ i, ref (a/a ref ) 3(+ω i), (2.22) with κ B (a) i ϱ i(a)/ M pl 2 H2 ref as well as one equation that links a and t, namely ȧ = H ref h(x(a), y(a), a) a. The index ref describes some reference point where the initial conditions x(a ref ), y(a ref ) and the densities of each background component at the reference scale ϱ i,ref are fixed. Typically, it will be after the end of inflation. The form (2.9) is especially suitable for numeric calculations since it only contains dimensionless normalized variables. All relevant quantities can easily be retrieved from the solutions x(τ) and y(τ), e.g. Ω φ = (x 2 /2 + U(y))/h(x, y, a) 2. It is even possible not to keep only φ and φ (i.e. y and x) as independent dynamic variables but also to switch to Ω φ and ω φ if the potential V (φ) is bijective: [ dω φ = Ω φ ( Ω φ ) 3( + ω φ ) + d ln κ ] B(a), (2.23a) dτ dτ dω φ dτ = 3( ω 2 φ ) ( ( )) 2U U Ω φ κ B (a) + ω φ ( ω φ ) 2 Ω φ Ω φ Ω φ κ B (a). (2.23b)

15 CHAPTER 2. COSMOLOGY AND QUINTESSENCE These sets have been obtained by a straightforward calculation using the Friedman equations and the equation of motion for φ. All numerical calculations in this work are based on these forms. 2.3 Constraints on Dynamical Dark Energy To be able to judge the viability of solutions from the equations of motion of dynamic dark energy it is important to know which principle and observational restrictions have to be obeyed. The main difference between a cosmological constant and dynamic dark energy models is that there can be a non-negligible dark energy content even in the early phase of the universe, whereas a cosmological constant is totally unimportant at that time. Furthermore, the equation of state parameter can differ from in dynamical models. The basic constraint is that dark energy can only have become dominant recently, i.e. there is a cross-over where Ω DE overtakes Ω M [62]. Furthermore the equation of state parameter ω DE has to be close to today in order to be consistent with observations Dark Energy in the Early Universe The dark energy fraction Ω today DE 0.7 cannot always have been dominant. There are some important phases during cosmic evolution where observations constrain the dark energy content of the universe: Big Bang Nucleosynthesis: The explanation of the observed abundances of light elements depends on the interplay between the nuclear reaction rates and the expansion rate of the universe H. This constrains the total energy content of the universe during BBN giving an approximate upper bound varying from Ω BBN DE [28, 80] down to Ω BBN DE [7]. Cosmic Microwave Background: The shape of the CMB-spectrum depends on the expansion history of the universe. For example, the angular size of the anisotropies is sensitive to the time of last scattering and the age of the universe which is influenced by the dark energy dynamics. Therefore the bounds are model-dependent. Caldwell and Doran [5] investigate various Quintessence models using WMAP data getting roughly Ω CMB DE as an overall upper limit. Structure formation: Strong accelerated expansion would hinder structure formation. This gives an approximate bound Ω SF DE [27, 28] Supernovae type a: The most recent constraint (z < 2) from the measurement of the luminosity of distant thermonuclear explosions of white dwarfs [62] gives among other results 9 Ω today DE One can even retrieve information about the recent evolution, i.e. ω DE. This is discussed in the next section.

16 CONSTRAINTS ON DYNAMICAL DE 0,9 0,8 Ω Λ Ω φ (LKT) Ω DE 0,7 0,6 0,5 SNe Ia 0,4 0,3 0,2 0, Structure CMB BBN e+05 e+09 e+0 e+ z+ Figure 2.2: Constraints on early dark energy from SNa, CMB, BBN and structure formation. The dashed line is the cosmological constant, the solid line is a leaping kinetic term Quintessence model with exponential potential. [28] Equation of State SN a observations can probe recent (z 2) dynamics of dark energy. Because the measured angular distribution of the CMB also depends on the whole expansion history since last scattering, it is also sensitive to the more recent evolution of dark energy. Similar is true for the measured large scale structure (LSS) power spectrum (see [8] for a detailed discussion). One can therefore extract information about the evolution of dark energy, expressed by the equation of state parameter ω DE, from the observational data. However, this requires assumptions about other cosmological parameters and the functional form of ω DE (z) as well as its allowed range (i.e. values smaller than ). For example, hardly any information could be retrieved without fixing Ω total = (see [74] for details). A first approximation would be to assume a constant ω DE in the recent past. Using a linear parameterization ω DE = ω 0 + ω z goes one step further. All bounds for ω DE are compatible with for the cosmological constant. It could even be smaller than (which is not possible in minimal Quintessence theories). This would mean that either the dominant energy condition ϱ p > 0 is violated 0, leading to superluminal velocities and an increasing dark energy density ϱ DE (t), or that the framework considered is incomplete and gravity has to be modified. The upper bound on the equation of state parameter is roughly ω today DE 0.7, (2.24) 0 Actually the dominant energy condition is only violated if the total density is smaller than the absolute value of the total pressure, which means a value ω DE ( + Ω M /Ω DE).4 would be allowed. However the energy density of a phantom with ω DE < increases, meaning that Ω M /Ω DE 0 quite fast. Thus the dominant energy condition will be violated sooner or later or ω DE has to increase rapidly in the near future. The weak energy condition ϱ + p 0 is also violated.

17 CHAPTER 2. COSMOLOGY AND QUINTESSENCE SNe Ia Ω M=0.27+/ dFGRS WMAPext 99% 95% 99% 68% ω DE % 68% % 95% Ω M 99% Ω M Figure 2.3: Left: Likelihood contours in the (ω DE, Ω M )-plane (the index M stands for pressureless matter) for static equation of state using the gold SNa sample of Riess et.al. with a prior on Ω M (left panel) and combined with CMB and LSS data (right panel) [62]. Right: Likelihood contours in the (ω 0, ω )-plane for a linear parameterization ω DE = ω 0 + ω z with a prior on Ω M and the gold SNa sample. The dot is the cosmological constant [62]. which is a very conservative limit consistent with many static and dynamic analyses for the equation of state parameter [5, 33, 62] (see left side of figure 2.3). The observational uncertainty for the measurement of the evolution of the equation of state parameter is still too large and needs too many assumptions about uncertain parameters (e.g. the Hubble constant) to be able to use it profitably for the limitation of dark energy models (see right part of figure 2.3). This may change soon, giving the possibility to distinguish observationally between some classes of dark energy models (see e.g. [65, 66]) Inhomogeneities The dark energy content of the universe is by definition a homogeneously distributed component on scales which are accessible by observations (an upper limit is the Hubble radius). However, if dark energy is described by a field it could in principle vary not only with time but also in space. There are several sources of inhomogeneities that have to be either ruled out or investigated: Gravitational Jeans-instability (similar to ordinary matter), Direct coupling to matter could transmit inhomogeneities in the matter distribution ϱ M (x) to the dark energy field, Inhomogeneities in the metric g µν (x) (due to an inhomogeneous matter distribution) could imprint themselves on dark energy due to gravitational coupling. In case of Quintessence, the interplay of the fluctuations in the field φ(t, x) = Φ(t) + ϕ(t, x) (where Φ(t) is the homogeneous background solution) with fluctuations in the metric ds 2 = dt 2 a(t) 2 (δ ij h ij (t, x))dx i dx j and the matter distribution ϱ M (t, x) = The h ij describe the deviation from the Robertson-Walker-Metric in the so-called synchronous gauge

18 CONSTRAINTS ON DYNAMICAL DE ϱ M (t)[+δ(t, x)] have to be studied. This was done in great detail already in one of the first papers about Quintessence by Rhatra and Peebles [60] and is beyond the scope of this work. However, using the approximation of the equation of motion (2.6) for φ(t, x) in first order in the fluctuations (in momentum space and with h tr h ij ) one gets: ϕ k + 3H ϕ k + (V (Φ) + k 2 /a 2 )ϕ k = 2ḣ Φ. (2.25) It is easy to see that, as long as the potential is convex, there cannot be a Jeansinstability since then V (Φ) + k 2 /a 2 is always positive. A direct coupling of Quintessence to baryonic matter has not been investigated by Peebles since it has to be extremely suppressed (see section 3.4.). Note, however, that this could be circumvented by a coupling to dark matter. Actually, the most stringent bound for such a coupling comes from its influence on structure formation [2]. Furthermore, it turns out [60] that a discrimination between super and sub horizon fluctuations is necessary. To fulfill the criterion of homogeneity inside the horizon the dynamic mass m φ V (Φ) /2 typically has to be of the order of the inverse size of the horizon given by the Hubble parameter H. This is typically the case in Quintessence models with tracking solutions (see section 3.3).

19 Chapter 3 Quintessence Dynamics In this chapter some common dynamic dark energy models will be presented and discussed. They are based on the Quintessence action (2.2) that just enlarges the usual action in curved space-time by a scalar-field φ in a straightforward way with standard kinetic term and a potential V (φ). In general, to get a solution of the cosmological smallness problem, the energy density ϱ φ given by (2.5) should decrease from very high values (up to M 4 pl ) to the small value it has today (typically of order ϱ c = H 2 0 M 2 pl ) during the evolution of the universe. This can be achieved by a decreasing potential in which the Quintessence field rolls down (see figure 2.). Furthermore, it is desirable that the equations of motion allow attractive solutions, which guarantees that the same behavior is exhibited for a broad range of initial conditions of φ and φ during the early universe thereby eliminating any fine-tuning once a specific potential is chosen. The further outline is to discuss models with exponential potentials first, in order to demonstrate the important issues like attractors in phase space and impact on the cosmological problems defined in section (2.). After that, alternative models and a more general discussion of the properties of V and the tracking solutions will be presented. At the end, extensions of the Quintessence action will be discussed. 3. Models with Exponential Potential The basic properties of a cosmological scalar-field with an exponential potential will be discussed first. After that, viable models formed from several exponentials will be explained. 3.. Single Exponential Actually, an exponential potential has already been discussed in the first papers about Quintessence by Peebles and Rhatra [60] who reconstructed it from a desired scaling solution and by Wetterich who motivated it by conformal anomaly [80] (see section 3.5.2). It is convenient to use a parameterization ( ) V (φ) = M pl 4 exp λ φ Mpl (3.) with the reduced Planck mass Mpl = 3/8πG and one free parameter λ. The scalarfield will roll down the potential, thereby going to higher and higher field values φ. There are many ways to obtain the well-known solutions for φ(t) in this potential.

20 6 3.. EXPONENTIAL POTENTIALS Here an illustrative possibility will be presented, using the dark energy fraction Ω φ and equation of state parameter ω φ as independent variables. All results are of course wellknown [6]. The general form of the equations of motion has been reported in (2.23). For the exponential potential (3.) and with only one dominant background (i.e. matter or radiation) component with constant equation of state parameter ω B they read: dω φ dτ dω φ dτ = Ω φ ( Ω φ ) [3(ω φ ω B )], (3.2a) ] = ( ωφ [3 2 ) λ Ω φ /( + ω φ ). (3.2b) The two-dimensional phase-space can be investigated by finding the nullclines: dω φ dτ = 0 Ω φ = 0 or Ω φ = or ω φ = ω B, (3.3a) ( ) dω φ 3 2 dτ = 0 ω φ = or ω φ = + or Ω φ = ( + ω φ ). (3.3b) λ The first two nullclines of each equation guarantee that the variables will stay in their allowed range 0 Ω φ and ω φ respectively. One possibility for the nullclines is given in the left upper panel of figure 3.. Fixed points in the (ω φ, Ω φ )- phase-space are given by intersections of two nullclines from (3.3a) and (3.3b). The stability of the fixed points can be calculated from a linearization of the system of equations (3.2) and determination of the eigenvalues of the corresponding 2 2 matrix. Alternatively, the stability can easily be inferred from the phase portraits like in figure 3.. There are four fixed points at the edges of the physical phase-space which are never stable for the relevant parameter range < ω B < + and λ > 0. Imagining that the equation of state parameter of the dominant background component ω B has been fixed, there are basically two different scenarios depending on the slope ( 3 2 λ) of the last nullcline in (3.3b): Dominant dark energy attractor: For λ < 3 + ω B the slope of the nullcline is so large that there is no intersection with the (vertical) ω φ = ω B -nullcline inside the physical range of parameters (see lower right panel of figure 3.). The only attractive fixed point is the intersection with the (horizontal) Ω φ = -nullcline: Ω φ = ω φ = ( λ 3 ) 2. (3.4) The equation of state parameter is constant and ωφ < ω B which means that the Quintessence energy density decays slower than the background energy density. Thus the dark energy dominates in the end leading to Ω φ = when the fixed point is reached. For λ 0 the potential gets flat and the field behaves like a cosmological constant, i.e. ωφ. Tracking attractor: For λ 3 + ω B there is an intersection with the vertical ω φ = ω B -nullcline (see lower left and upper right panel of figure 3.) leading to an attractor ( ) 3 2 Ω φ = ( + ω B ) ωφ λ = ω B. (3.5) In case of λ < 0 one can define a new field φ = φ. It is completely equivalent to use either φ and a potential with λ or φ and λ. Therefore only λ > 0 will be considered here.

21 CHAPTER 3. QUINTESSENCE DYNAMICS 7 Figure 3.: Phase portraits in the (ω φ, Ω φ )-plane for different values of ω B and λ. Upper row: phase-space with nullclines for ω B = /3 and λ = 5 (left, nullclines for Ω φ are blue and red for ω φ, the unpysical regions of phase-space are shaded grey) and with phase-space trajectories (right, the green line is a possible trajectory into the fixed point). Lower row: phase-portraits for ω B = 0, λ = 5 (left) and ω B = 0, λ = 2.5 (right). All plots have been made using xpp. The equation of state parameter is constant and equal to the one of the dominant background component. That means that the cosmological evolution of the Quintessence energy is proportional to the scaling of the energy density of the dominant background component ϱ φ (t) ϱ B (t), i.e. Quintessence tracks the background. This leads to a constant fraction 2 ϱ φ /(ϱ φ + ϱ B ) Ω φ which is also determined by the parameter λ and can range from (for λ = 3 + ω B ) to 0 (for λ ). For 3 + ω B λ 3 2 the dominant dark energy fixed point still exists but is not attractive any more. It vanishes totally when the slope 9/λ 2 of the nullcline gets smaller than /2 (for λ > 3 2). The behavior of the tracking attractor is very interesting in two respects: First, it is an example where the Quintessence energy decreases dynamically thereby indeed solving the cosmological smallness problem (this was the major motivation for dynamic dark energy models). Second, after a phase of transient oscillations until the fixpoint is approached, the Quintessence energy directly follows the density of radiation and later (after the beginning of matter domination) the density of matter in cosmic evolution (see figure 3.3). The energy density of Quintessence and the background are compa- 2 Remember that there is no curvature contribution since k = 0 has been chosen.

22 8 3.. EXPONENTIAL POTENTIALS rable during the whole history of the universe. This can be considered as a natural explanation of the coincidence of scales in this model. Furthermore, there is only one attractive fixed point in phase-space. All trajectories end in this fixed point ( maximal basin of attraction ). Thus dependence on initial conditions is completely wiped out and no fine tuning is necessary. It is also very interesting to calculate the squared dynamic mass V for the two attractors. 9( ωb 2 ) H 2 for λ > 3 + ω B V = λ (2 λ2 9 ) H 2 for λ < 3 + ω B (3.6) The attractors therefore both fulfill the condition that the dynamic mass is always of the order of the inverse size of the horizon as was discussed in section However, neither of the two attractors alone provides a viable model: If the tracking attractor is realized, the constant fraction 3 Ω φ can be chosen (by fixing λ) either close to 0.7 (for λ 3.6) or smaller than 0. (for λ ) as required by the BBN bound from section (2.3.), but both requirements cannot be fulfilled simultaneously. Furthermore ω φ would be zero today (since ω B = 0 for matter domination) leading to no acceleration. If the attractor with dominant dark energy is realized, then Ω φ = as soon as the attractor is reached, which is obviously also not acceptable from an observational point of view 4. One way how to solve this puzzle without loosing the nice properties of the attractors will be discussed in the following section Multiple Exponentials A tempting way how to use the appealing features of the two attractors in the exponential potential and to fulfill the observational bounds on the dark energy content during the evolution of the universe (see section 2.3.) would be to combine the two attractors to a cross-over Quintessence model: In the early universe the tracking attractor is realized. The Quintessence energy tracks the background energy. The cosmological smallness problem is solved and the coincidence of scales is explained through the tracking. To fulfill the upper bound Ω φ 0. during BBN and CMB any exponential with λ (3.7) is fine (see 3.5). Due to the attractor property no fine tuning is necessary. In the present cosmological epoch, the dominant dark energy attractor is being reached. Ω φ will increase and will reach the value of the fixed point in the future. Simultaneously, ω φ will change from ω B = 0 to ωφ = + λ2 /9 in the future. With matter as dominant background this is only possible if (compare with assumptions for (3.4)) λ < 3. (3.8) 3 Actually, Ω φ is constant during radiation domination, then decreases by a factor 3/4 (due to the factor + ω B in (3.5)), and is constant again during matter domination. However, this does not change the basic argument. 4 This could be circumvented by fine-tuning the initial conditions in such a way that the attractor is reached just today. However, this seems quite unsatisfactory from a theoretical point of view.

23 CHAPTER 3. QUINTESSENCE DYNAMICS 9 Furthermore, in a realistic model λ has to be even smaller in order to be able to fulfill the bound ω φ < 0.7 (see (2.24) and right panel in figure 3.2)). Obviously the single exponential potential (3.) cannot accommodate this scenario since two distinct values for λ are required. An effective change in the slope of the exponential potential parameterized by λ can be reached in various ways: Add exponentials with different slope. The simplest case is: V (φ) = M 4 pl [ ( exp λ ) ( )] φ φ + B + exp λ 2. (3.9) M pl M pl The exponential with larger λ (e.g. λ ) will dominate for small φ, the one with smaller λ (e.g. λ 2 ) for large φ (see [6] and figure 3.3). Replace λ by a function λ(φ) that smoothly varies between the two desired values. Keep the potential as it is and introduce a kinetic term k(φ) 2 ( φ) 2 with a field normalization k(φ) that smoothly varies between two values. ( leaping kinetic term, [37]). This is equivalent to the previous item via a field redefinition φ k(φ)dφ. Albrecht and Skordis [] find viable solutions by considering exponential potentials with φ-dependent prefactors of the form [a + (b φ) α ]. Exemplarily, the first case will be discussed. It can be shown [6] that viable solutions can be found for a large parameter range inside the allowed region (λ, λ 2 < 3) (see right panel of figure 3.2, the choice of B will be discussed in a moment). Ω φ Ω φ and ω φ today λ =2, λ 2 =2.0 λ =2, λ 2 =.5 λ =2, λ 2 =.0 λ =2, λ 2 = ω φ λ BBN/CMB excluded SNa excluded ω φ = 0.6 ω φ = 0.7 ω φ = 0.8 ω φ = 0.9 ω φ = λ Figure 3.2: Left: Evolution of (ω φ, Ω φ ) for λ = 2 and various λ 2 = 0.5,,.5, 2 (from left to right). The system is first in the tracking fixed point (0, 9/λ 2 = 0.06). In the future it approaches the dominant dark energy fixed point ( + λ 2 2 /9, ). We are living in the transient period where the system moves along the displayed trajectories from one fixed point to the other. The green area is the allowed parameter range today. Any realistic model has to cross this area. Right: Contour plot of today s equation of state parameter ω φ for different values for λ and λ 2. Large λ and small λ 2 leads to values for ω φ close to. The bound (3.7) from Nucleosynthesis (BBN) and CMB is shown as well as the bound ω φ < 0.7 (2.24) from SNa. The remaining parameter space can yield viable models. Both plots have been done based on xpp calculations. A similar plot to the right one can also be found in [6].

24 EXPONENTIAL POTENTIALS According to this model, we are living in the transient period where the field switches from the tracking attractor with Ω φ = const 0. to the attractor were Quintessence dominates Ω φ = with a present value Ω φ 0.7 (see left panel of figure 3.2). The fact Figure 3.3: Schematic sketch of the double exponential scenario. Left: Tracking occurs during radiation- and matter domination. The present change of the behavior of the Quintessence field is due to a flattening of the potential (right). The potential is given by (3.9) with V = M 4 pl and V 2 = M 4 pl e λ2b/ M pl [7]. that our present cosmological epoch is special (in the sense that expansion switches from decelerated to accelerated, coincidence of epochs ) is explained in this model by a change in the slope of the Quintessence potential that happens for φ(t) φ today (see figure 3.3). The price that one has to pay for this is that the new free parameter B in the potential (3.9) has to be tuned in such a way that the change of the potential happens approximately today which turns out to be very roughly of the order B O(φ today λ λ 2 λ 2 ) O(00M pl ) depending on λ and λ 2 [6]. Discussion One can adopt two (extreme) points of view about the choice of the parameter B: It has actually been measured by SNa that acceleration is a recent phenomenon. This is reflected by the potential, and the parameter B is determined by a measurement. All other parameters have natural values, i.e. neither extremely large (in Planck units) nor extremely small. The parameter B has to be fine-tuned over two orders of magnitude to accommodate the coincidence of epochs. This is just a rewriting of the usual fine-tuning problem (which has to be done over 20 orders of magnitude) by tuning the exponent in the potential. Actually, the same problem arises in the leaping kinetic term scenario. However, if the transition between the two attractors was indeed triggered by a change in the field normalization the tuning over two orders of magnitude, which is still necessary, would not appear in the exponent. It is stressed e.g. in [37] that there is no reason why the field normalization should stay constant during the whole history of the universe. The transition in the field normalization could be linked with some present procedures (like structure formation) [37].

25 CHAPTER 3. QUINTESSENCE DYNAMICS 2 Anyway, it would be a great achievement if some natural explanation for the transition between the two attractors could been found which otherwise seems, in our opinion, somehow artificial. The advantages of the models presented here are that the cosmological smallness problem is resolved and the coincidence of scales is naturally embedded in this kind of models, which is not the case for a cosmological constant. Furthermore, the exponential potential is a very good tracker, i.e. trajectories corresponding to many initial conditions can reach the fixed point in a time shorter than the age of the universe [6]. 3.2 Models with Inverse Power Law Potential Besides the exponentials, another important class of Quintessence potentials are the inverse power laws often parameterized as [43, 60] V (φ) = Λ 4+α /φ α (3.0) with a parameter Λ of mass dimension one and the power law index α Attractors of Inverse Power Laws Like in the case of exponentials these potentials possess attractive solutions. One basic difference to the exponential case is that one has to differentiate between two cases: scalar-field dominated cosmology and background dominated cosmology. Background dominated cosmology and the tracking attractor In the latter case it turns out that the subdominant Quintessence field also tracks the dominant background. The corresponding attractive solution in phase space can easily be found by making an ansatz φ(t) t β (3.) in the equation of motion (2.2). The Friedman equation for H (2.9a) decouples from the scalar-field since ϱ B ϱ φ and yields the usual solution H = 2 3( + ω B ) t η t (3.2) in case of a background which is dominated by one fluid with equation of state parameter ω B. A simple calculation shows that β = 2/(α + 2) and φ(t) = ( 9( ω 2 φ ) 2α 2 ) α+2 H 2 α+2 Λ α+4 α+2 (3.3) which is an attractive solution with constant equation of state given by [43, 60] ωφ = + ( + ω α B) α + 2. (3.4)

26 INVERSE POWER LAW POTENTIAL Actually the previous result can easily be obtained by calculating the scaling of the energy density using φ(t) and a(t) t η yielding 5 ( ) 2 4+α ϱ φ Λ 2 4+α 2+α t 2α α+2 M pl 4 2+α a 3(+ω B ) Λ Mpl α α+2 a 3(+ω φ ) (3.5a) ϱ B M 2 pl t 2 M 4 pl a 3(+ω B) (3.5b) which has been contrasted by the corresponding standard result for the scaling of the background. Using (3.3) to calculate the squared dynamical mass V one finds V (φ(t)) = 9 2 α + α ( ω φ 2 ) H 2, (3.6) which is in agreement with the condition discussed in section For α > 0 the energy density of the Quintessence field decays slower than the background (i.e. ωφ < ω B). Furthermore, in order to fulfill the condition that the background dominates the energy density it is obviously necessary that Λ M pl. In this case, this attractor tracks the background: During radiation domination: ϱ B a 4 and ϱ φ a 4 α α+2, During matter domination: ϱ B a 3 and ϱ φ a 3 α α+2. Like in the case of an exponential tracking attractor, the dark energy density decays during cosmic evolution, thus solving the cosmological smallness problem (which was the main motivation for such dynamical models). However, the ratio ϱ φ /ϱ B is not constant (like for the tracking in an exponential potential) but increases more and more (as soon as the attractor is joined). At some time, the ratio ϱ φ /ϱ B will therefore approach unity, depending on the parameter Λ in the potential. The approximation of background domination cannot be used any more in that case. The two densities will perform a cross-over thereby entering the period of Quintessence dominated cosmology. Quintessence dominated cosmology After the Quintessence energy has overtaken the background energy, which happens very naturally as described above, the system settles down in another attractor [43, 60]. The dominating Quintessence energy enters into the Hubble parameter (first Friedman equation (2.9a)) which acts as a friction term in the equation of motion (2.2) of the Quintessence field, slowing down the decay of the Quintessence field. This retardation, on the other hand, leads to a very slowly rolling field with an equation of state parameter that approaches in the future. The expansion of the universe accelerates. Using the equation of motion (2.2) and the Friedman equation (2.9a) with an analog scaling ansatz as in (3.) together with the slow roll conditions φ 2 V and φ 3H φ, V 5 The normalization of a(t) is arbitrary. Here it has been chosen in a convenient way to compare Quintessence and background energy densities.

27 CHAPTER 3. QUINTESSENCE DYNAMICS 23 and of course ϱ φ ϱ B yields [60] ( ) 2 t α+4 φ(t) = c Λ t pl 2 ω φ (t) = + φ(t) + c 2 2 a(t) exp ( c 3 φ(t) 2 M 2 pl M 2 pl ) (3.7a) (3.7b) (3.7c) with c = (α(α + 2)/6) 2/(4+α), c 2 = (α + 4) 2 /2c α+3, c 3 = (α + 4)/2(α + 2)c (α+2)/2 and t pl = / M pl. It can be seen that (despite the slow roll) φ for large t, and therefore ω φ. The universe will stay in the dark energy dominating attractor and inflate forever Dark Energy Model with Inverse Power Laws In contrast to the exponential potential, one single inverse power law term can accommodate a model that exhibits the desired behavior of a cross-over dark energy model. The transition from a decelerated to an accelerated expansion happens during the transient epoch where the system evolves from one attractor to the other, which is very similar to the models with exponential potentials. The cross-over is not generically linked with our present cosmological epoch, but depends on Λ: It will happen when the Quintessence density (3.5a) and the background density (3.5b) are getting of the same order of magnitude: cross over Λ 2 4+α 2+α t 2α α+2 M 2 pl t 2. (3.8) In order to make the cross-over happen in the present epoch, Λ has to be chosen in such a way that the above relation holds for t t 0 /H 0 (where t 0 is the present age of the universe). This rough order of magnitude estimate yields Λ = O ( ( H0 M pl ) 2 4+α Mpl ) ) = O ( α Mpl. (3.9) Note that this choice implies that the field φ(t 0 ) is of the order M pl today (see (3.3)). α O(Λ) 0 mev 0 ev 0 kev 0 2 MeV 0 2 GeV 0 4 GeV 0 7 GeV Table 3.: The order of magnitude of Λ for potentials with different power law indices. Depending on α, the required value of Λ lies in a region of typical standard model and high energy scales (see table 3.). Once the parameter Λ is fixed, the coincidence of scales and the coincidence of epochs are explained by this model.

28 INVERSE POWER LAW POTENTIAL Basin of attraction The appealing property of attractors to wipe out dependence on initial conditions requires that many trajectories in phase space are drawn towards the attractive solution. One can define the range of initial conditions from which the attractor is reached as its basin of attraction. However, for practical purposes it is necessary that the attractor is reached in a time shorter than the age of the universe (starting at the time when initial conditions are fixed). The range of initial conditions for which this is the case can be considered as an effective basin of attraction which can potentially be smaller than the full basin of attraction 6. First it has to be discussed when, during cosmic evolution, the initial conditions are fixed. The general view is that this happens directly after inflation [70]. However, if the Quintessence potential is formed later (e.g. when the temperature falls below Λ) the fixing of initial conditions will also take place at a later time. The basic scenario does not depend strongly on this issue, but nevertheless one should keep that aspect in mind. How the tracking attractor is approached has been investigated in great detail by [70], see also []. Basically, one can distinguish between two cases: The initial Quintessence energy ϱ φi can be higher than the value given by the attractive tracking solution (overshoot) or it can be lower (undershoot). For simplicity, the discussion will be restricted to initial conditions where the field is released from rest ( φ = 0). In the case of an undershoot, the motion of the scalar-field is overdamped. It is basically frozen to its initial value and sits and waits until the tracking solution is reached (see figure 3.4). If the initial energy is too high, the equation of motion is dominated by the potential. The Quintessence field freely rolls down ( fast roll with ω φ = + and ϱ φ a 6 ) the potential. The Quintessence energy decays fast and at some point the tracking solution is reached. However, the kinetic energy is too high and the field overshoots the tracking attractor (see figure 3.4). As soon as this has happened the scalar-field is slowed down. The kinetic energy decreases and at some point the same happens as in the undershoot scenario: The field freezes until the tracking solution is reached again and finally joined. Initial conditions φ 0 do not change the behavior significantly [70]. If the initial kinetic energy is very large, the field will start with a fast roll like in the overshoot case. One can now ask: What are the maximal and minimal initial energy densities for which the attractor is reached just today? The minimum is clearly given by today s critical density ϱ c0. In this extreme undershoot case the Quintessence energy stays constant and crawls into the tracking solution just today. The maximal overshoot is given by the condition that ϱ φ freezes on a level not lower than ϱ c0. Maximum and minimum are shown in figure 3.4 for several values of α. For steep potentials (α ), the basin of attraction spans an enormous range of up to 00 orders of magnitude 7 [70] including equipartition initial conditions Ω φ /#dof 0 3. On the other hand, shallow potentials (small α) are poor trackers with small basin of attraction. In the extreme case α 0 the potential gets flat and the cosmological 6 In future, the term basin of attraction will be used for the effective one. 7 Note that the actual size of the basin of attraction depends on the time when initial conditions are fixed. If inflation ends early and initial conditions are fixed immediately thereafter the basin of attraction can be much larger than displayed in figure 3.4. However, it could be also vice-versa.

29 CHAPTER 3. QUINTESSENCE DYNAMICS 25 Figure 3.4: Evolution of the Quintessence energy for different α and initial conditions. The tracking solution is red, the background energy density is blue and the various black curves correspond to over- and undershoot initial conditions including the minimal and maximal case. The density on the y-axis is given relative to the initial background density ϱ Bi. The scalefactor is normalized to when initial conditions are fixed. Today corresponds to the point just after the equality of background and Quintessence energies. The fixing of initial conditions has been chosen to happen for a redshift of about z 0 20 as an example (i.e. log a 20 today). If this value is increased (or decreased), the basin of attraction will grow (or shrink) following the maximal and minimal trajectories. Plots have been done using xpp, see also []. constant is reproduced. In this limit, the basin of attraction shrinks to a point (equal to ϱ Λ ) to which the initial condition has to be fine-tuned. Equation of state Inverse power law models predict that the Quintessence field changes from the tracking attractor (ω φ = + α α+2, Ω φ ) (for ω B = 0, see (3.4)) towards the dark energy dominated attractor (ω φ, Ω φ = ) in the present cosmological epoch. It is an important question whether the transition is fast enough in order to fulfill the bound ω φ 0.7 today (i.e. for Ω φ 0.7, see (2.24)). In contrast to the double exponential potential (3.9) there is only one free parameter α here. A numerical analysis shows that the bound on ω φ can only be fulfilled for small values [26] (see also figure 3.5) α 2 (3.20) or even smaller [] depending on the observational data and method of analysis.

30 TRACKING SOLUTIONS Ω φ Ω φ and ω φ today α=6 α= α=0.5 α= Figure 3.5: Evolution in the (ω φ, Ω φ )-plane for different α from the tracking attractor ( + α α+2, 0) to the dark energy dominating attractor approaching (, ). The green area are bounds for today s values. Any realistic model has to hit this area. Plots made using xpp. ω φ Discussion The inverse power law potential is one of the archetype Quintessence potentials [60] that shows quite naturally many desired properties. Attractors abandon fine-tuning once a potential is fixed and the interplay of the two attractors for the backgroundand Quintessence dominated regimes naturally assure that a phase with negligible Quintessence content is followed by a phase in which dark energy dominates and mimics the equation of state of a cosmological constant. This behavior solves the cosmological smallness problem. The coincidence problems are also solved if one accepts the choice of the parameter Λ given by (3.9). The need to fix Λ at a certain value expresses the fact that, despite the cross-over happening naturally, it is not generically linked with the present epoch. Analogically to the exponential potentials, one can consider the choice of Λ either as unavoidable tuning or as the determination of a free parameter by a measurement. Anyway, the required order of magnitude of Λ inspires the search for some high energy physics explanation of inverse power law potentials (see section 3.5. and [9]). However, a closer look shows that the simple inverse power law runs into problems since a good tracking (large basin of attraction) in the early universe and a rapid crossover (ω φ < 0.7) today require contradicting values of α. Note that these two issues are not linked in principle, but only for this specific type of potentials. The problem can easily be circumvented by considering more general potentials, like sums of inverse power law terms (see section 3.3.4). 3.3 Tracking Solutions The two types of potentials discussed so far showed many similarities with respect to their basic shape (e.g. V (φ) 0 for φ ) as well as with respect to their solutions (e.g. attractors with constant equation of state). This motivates the question whether one can formulate very general conditions for potentials V (φ) that can yield viable models The Tracker Condition The notion of tracker fields has been introduced by Steinhardt, Wang and Zlatev and the argumentation here mainly follows [70]. Starting point is to define a tracker

31 CHAPTER 3. QUINTESSENCE DYNAMICS 27 solution by some desired properties and then try to find criteria whether a given potential V has such solutions or not. The desired properties are:. Attractor solution: Convergence from many initial conditions until now. 2. Tracking of the background: The background (characterized by ω B ) tells the Quintessence field how to behave, i.e. ω φ is a function of ω B and (nearly) constant. 3. Cross-over: For ω φ < ω B the ratio ϱ φ /ϱ B (which should be small in the past) increases with time leading naturally to a cross-over and a transition to a dark energy dominating attractor. From property, one can conclude that the properties of the desired solution only depend on the potential since dependence on initial conditions is washed out. Steinhardt, Zlatev and Wang [70] now rewrite the equation of motion (2.2) using the Friedman equation (2.9a) in the following ways: V V = [ 9( + ω φ ) + ] M pl Ω φ 6ẋ (3.2a) V V V 2 = + ω B ω φ 2( + ω φ ) + ω B 2ω φ ẋ 2( + ω φ ) 6 + ẋ 2 ẍ + ω φ ( + ω φ ) 2 (3.2b) where x ( + ω φ )/( ω φ ), ẋ d ln x/d ln a and ẍ d 2 ln x/d(ln a) 2. The left hand side of the second line is called Γ V V/V 2. Using property 2, one obtains ẋ and therefore the first equation (3.2a) implies the so-called tracker condition V V Ω φ (3.22) which has to be fulfilled during the evolution of the universe. Using property 3 shows that Ω φ should be small but increasing with time. Thus the tracker condition yields 8 0 < d V dφ V = V V V 2 V 2 = V 2 ( V ) V V 2 V 2 = V 2 (Γ ) (3.23) V 2 which means that Γ >. Using again ẋ and ẍ (from property 2) in equation (3.2b) one can see that Γ is approximately constant if ω φ fulfills property 2. These results motivate the following theorem [70]: Tracking behavior (i.e. a solution that fulfills properties - 3) occurs in any potential for which Γ > and Γ is approximately constant over the range of plausible initial conditions. (3.24) The full proof (especially of the attractivity) can be found in [70]. Plausible initial conditions means that φ can vary such that V (φ) lies between today s critical density ϱ c0 and the background density ϱ Bi when initial conditions are fixed. With this theorem, one can judge whether tracking solutions exist for a given potential V without 8 It is assumed that φ rolls down its potential, and therefore sgn(dφ/dt) = sgnv.

32 TRACKING SOLUTIONS solving the equations of motion, but only by calculating some simple derivatives of the potential. The condition that Γ has to be approximately constant can also be quantified just using V [70] Γ Γ(V /V ), (3.25) which is equivalent to ẋ, ẍ. The condition can also be formulated as the requirement that Γ/Γ where Γ is the variation of Γ over the relevant range of φ. With the upper constraint it is even possible to calculate the value of the equation of state ωφ for the tracker solution just from the value of Γ by setting ẋ = ẍ = 0 in (3.2b) yielding (with ωφ ω φ for this case) ω φ = + + ω B 2(Γ ) +. (3.26) This is actually a very useful result since the most important property of the tracking solution, the equation of state parameter, can be obtained directly from the simple function Γ. Furthermore it turns out that potentials with ω B 6+2ω B < Γ < also possess attractive solutions that fulfill property and 2. However, they cannot explain a cross-over since the attractive solution has an equation of state parameter ω B < ω φ < ( + ω B )/2 and therefore ϱ φ /ϱ B decreases with time. For even smaller values Γ < ω B 6+2ω B there are no stable solutions (with nearly constant ω φ ) at all (see [70]) Dynamic Mass Scale For tracker solutions (with nearly constant equation of state ωφ ) one can find a simple relation for the dynamic Quintessence mass scale m φ V /2 using just the equations (3.2) and ẋ, ẍ yielding V = 9 2 Γ ( ω φ 2 ) H 2. (3.27) Similar equations have actually already been shown for the tracking solutions in exponential (3.6) and inverse power law potentials (3.6). The fact that the dynamic mass generally is of the order of the inverse size of the horizon given by H has also been shown to be necessary for the spatial homogeneity of the Quintessence field (see section 2.3.3). Thus, it is very reassuring that this property is always given for tracking solutions. This remarkable property can actually be understood as the result of an interplay between friction force 3H φ and potential force dv/dφ in the equation of motion (2.2). The relevant timescale for the damping is t fric /H whereas the typical timescale for the dynamic evolution (without damping) is given by t dyn (V ) /2 = m φ. It is instructive to consider two extreme cases qualitatively in a tracking Quintessence potential (i.e. with Γ const > and background domination leading to H /t): For t fric t dyn the system is overdamped and the field will freeze after the kinetic energy is dissipated away. Since t fric = /H t the dynamic scale t dyn (which accounts for shifts in the φ-field due to the gradient of the potential)

33 CHAPTER 3. QUINTESSENCE DYNAMICS 29 is much larger than the corresponding age of the universe t. Thus the field stays frozen even when φ gets so small that 3H φ < V. For t fric t dyn the field will fall down its potential and be rapidly pushed in a region where 9 t fric t dyn. However, the field has gained a huge kinetic energy φ 2 /2 V which then dissipates away (for kinetic energy domination the equation of state is ω φ = + and the kinetic energy decays quickly with a 6 ) until the potential energy becomes important again and the field behaves like in the overdamped case, i.e. freezes. Tracking can occur only for an intermediate case where both scales are similar 0, i.e. t fric t dyn or equivalently V H 2. Furthermore, even if one starts with very different orders of magnitude of the two scales, the system will adjust itself in such a way that they approach each other due to the fact that the damping decreases with time (H /t): The overdamped system will sit and wait (φ const, i.e. t dyn const) until the Hubble parameter H = /t fric /t has decreased so much that t fric t dyn and the freezing stops. The undamped system is first, like described above, pushed in an overdamped region and then also freezes (after it has lost its kinetic energy). Like for the overdamped system, the freezing will stop once t fric t dyn. This shows that the system naturally evolves towards solutions where the two scales are comparable t fric t dyn and thus can give an qualitative understanding why the Quintessence field typically possesses attractive solutions with V H 2. Of course the arguments above have to be understood as a rough sketch. For a quantitative treatment and stability analysis see [70]. Actually, the overdamped case corresponds to the undershoot scenario, the opposite case to the overshoot, as discussed for inverse power law potentials in section This behavior of approaching the attractor is typical for tracking solutions in general [70] The Prototype Potentials In this section the theorem (3.24) for tracking solutions will be discussed for the prototype potentials where the function Γ = V V/(V ) 2 is exactly constant. One can easily show that: V (φ) φ α ) for Γ > with Γ = + α exp ( λ φ Mpl for Γ = φ n for Γ < with Γ = n (3.28) 9 For potentials that allow tracking, V decreases with V (in the limit Γ const > ), and thus t dyn will increase when the field rolls down the potential. 0 In the extreme cases above one has either ϱ φ const (freezing) or ϱ φ a 6 (kinetic energy domination), whereas the tracking requires ϱ φ a 3(+ω φ ).

34 TRACKING SOLUTIONS The case Γ > Only for Γ > the theorem predicts tracking solutions with ω φ < ω B leading to a natural cross-over with all the desired properties discussed in section This class of potentials is given by the inverse power laws. All results are in agreement with the results from section 3.2. The case Γ = For the borderline case Γ = the theorem predicts ω φ = ω B (see (3.26)), which has indeed been obtained for the tracking attractor of the exponential potential in section 3.. Note that for Γ = the theorem does not make a statement about the stability of the tracking solution. Actually it has been shown that the exponential potential also possesses an attractor where dark energy dominates depending on the value of λ. This special feature has been used to construct viable models with multiple exponentials. In terms of Γ this means that Γ when only one exponential dominates the potential and Γ in the transient period where two exponentials are of the same order of magnitude. The case Γ < Finally, for Γ < there are also attractive solutions, however with ωφ > ω B. Thus potentials of the form φ n do not lead to an cross-over and an accelerated expansion. The stability condition Γ > ω B 6+2ω B translates to n > 6+2ω B ω B, i.e. n > 6 for matter domination and n > 0 for radiation domination. For smaller powers n there is no tracking with constant ω φ but the field will oscillate (damped) around the minimum (for even n). Neither case is suitable to accommodate a viable dark energy scenario. The only (obvious) way would be to add a constant V 0 to the potential which has to be tuned to exactly the observed value of dark energy. This is nothing else as the cosmological constant. Thus simple potentials, like quadratic or quartic, seem to be disfavored Alternative Potentials with varying Γ The great achievement of the tracking theorem (3.24) is that more general potentials, where Γ varies slightly over the relevant range of φ, can be considered. In the following two representative potentials of this class will be discussed. For a recent review see [65]. Multiple inverse power laws In section 3.2 it has been discussed that an inverse power law potential φ α with large α is favored from the point of view of the basin of attraction, whereas the constraint on today s equation of state implies small values α 2. Actually, this contradiction can be circumvented by considering sums of inverse power law terms [70] V (φ) = α c α φ α (3.29) where α runs over a finite or infinite set of positive integers. The function Γ will vary slightly between + /α max (or if α max = ) and + /α min. The theorem thus For two exponentials with parameters λ and λ 2 in the exponent the maximal value of Γ is + (λ λ 2) 2 /4λ λ 2 when both exponentials are of the same order of magnitude.

35 CHAPTER 3. QUINTESSENCE DYNAMICS 3 predicts a tracking solution with a slightly varying equation of state parameter given by (3.26) ending in a cross-over to the dark energy dominated regime. Steep contributions in the potential with large α will dominate in the early universe leading to a large basin of attraction. The more shallow terms in the potential with small α will start to dominate when the field value φ increases more and more. This makes it much more easy to fulfill the present bound on the equation of state, for which α 2 is needed. One example for an infinite sum of inverse power laws is given by the inverse exponential potential ( V (φ) = V 0 (exp κ M ) ) pl. (3.30) φ The function Γ is given by (with u φ/κ M pl ) Γ = ( + 2u)( e /u ) { for φ 0 2 for φ. (3.3) The limits correspond to α max = and α min = respectively. The tracking solution in this potential will start with (ω φ ω B, Ω φ ) for small field values corresponding to Γ, and then ω φ decreases adiabatically as Γ increases according to (3.26). At some time, the Quintessence content becomes non-negligible and the tracking attractor gets unstable. Like for the inverse power laws, the system will perform a cross-over and evolve towards a dark energy dominated attractor approaching (ω φ, Ω φ ). The evolution of the equation of state parameter during the cross-over (and before along the Ω φ = 0 - axis) can be seen in figure 3.6, in comparison with the (large α) tracker potential φ 6. Ω φ Ω φ and ω φ today κ=2 κ= κ=0.2 α= Figure 3.6: Comparison of the evolution in the (ω φ, Ω φ )-plane for the cross-over in inverse exponential potentials with different κ with an inverse power potential φ α with α = 6 (see fig. 3.5). The green area are bounds for today s values. Any realistic model has to hit this area. Plots made using xpp. ω φ Sugra potential Another often discussed potential is the so-called SUGRA-potential [3, 4] given by ( ) V (φ) = Λ 4+α φ α exp β φ2 M pl 2 (3.32) with Γ = + α + φ 2 /φ 2 m ( φ 2 /φ 2 m )2 and φ m α/2β M pl. (3.33)

36 EXTENDED QUINTESSENCE It is mainly equivalent to the inverse power law potential for small values φ M pl. This is the range of interest for the tracking solution. As Γ + α in this range the tracking will occur with approximately the same equation of state parameter ωφ as for the inverse power law (see (3.26) and (3.4)). The difference to the inverse power potential comes about when the cross-over happens for φ M pl. This type of potential actually has a minimum at φ = φ m in which the field will settle down leading to a cosmological constant like behavior with ω φ and V (φ) V min = (2eβ/α) α/2 Λ 4+α / M pl α. Λ has to be set to the right value to obtain today s dark energy density. Actually, this means that Λ is of the same order as for the pure inverse power laws (see table 3.), which means that the minimum is reached in the present cosmological epoch 2. The advantage of this potential with respect to pure inverse power laws is that high values of α can be chosen (with large basin of attraction) without spoiling today s bound for the equation of state for β (see figure 3.7). Ω φ Ω φ and ω φ today β=0.5, α=6 β=, α=6 β=2, α=6 α= Figure 3.7: Comparison of the evolution in the (ω φ, Ω φ )-plane for the cross-over in sugra potentials with α = 6 and different β with an pure inverse power potential φ α with α = 6. All models start at the same tracking attractor (ωφ = 0.25, 0) (see fig. 3.5) and only differ in their cross-over and late time behavior. Plots made using xpp. ω φ On the one hand, the introduction of a potential with a minimum of the order of today s critical density seems to be just a rewriting of the cosmological constant problem. On the other hand, all nice properties like dynamic solution of the cosmological smallness problem and independence of initial conditions are indeed realized in this model and no extraordinarily small scale has to be introduced in the potential. At least it is a viable example where the potential does not drop to zero for large φ values Extended Quintessence So far, the case of a cosmological scalar-field that is just coupled minimally to gravity by the action (2.2) and that has no interaction with the background Lagrangian of matter and radiation has been discussed. This is a very tense restriction, whatever origin the Quintessence field may have. 2 The choice of Λ in table 3. means that the tracking ends in the present epoch. Thus the field tracks until the recent past. 3 One could argue that it is always possible to take any Quintessence potential and modify it in such a way that it has a minimum that is just reached today. The evolution in the past will not be disturbed, and the minimum today leads to accelerated expansion. However, in general this means that some very small scales have to be introduced and fine-tuned in the potential. This is exactly what one wants to avoid.

37 CHAPTER 3. QUINTESSENCE DYNAMICS 33 In principle, there is a huge number of possibilities how a scalar-field could be coupled with geometrical quantities (like the curvature scalar R) or matter and gauge fields in the action. Since cosmological Quintessence models typically operate with highly nontrivial potentials V (φ), one generally has to take into account the possibility that an underlying theory will also produce nontrivial couplings. Furthermore, it will be shown later on, that quantum corrections induce an effective action that contains non-minimal coupling terms (see section 5.5). However, as long as one does not use a well-defined underlying theory for the Quintessence field (which may go hand in hand with an underlying theory of gravity) one cannot strictly calculate the form of the additional couplings. In this section some of the most prominent possibilities for extensions will be discussed Yukawa Coupling The simplest direct coupling of the Quintessence field to fermions is by Yukawa-type couplings [60]. A contribution to the Lagrangian could for example look like L Yuk = i F i (φ) ψ i ψ i, (3.34) where i runs over the different species of fermions 4. The functions F i of the Quintessence field can in principle, like the potential, be arbitrary functions of the Quintessence field φ. Each function F i (φ) gives a contribution to the mass (m i ) of each fermion. The fact that the field value φ(t) changes during cosmic evolution means that the fermion masses are also time-varying. Actually, this is a very typical feature of Quintessence models (see next section). Of course, the time-variation of the fermion mass is supposed to be tiny in comparison to the total mass. The fermions ψ i do not need to be fundamental fermions but should be understood as effective fields, e.g. describing neutrons or protons, with effective Yukawa couplings F i (φ). In this case, the φ-dependence of the nucleon masses could also be mediated by a φ-dependence of the QCD scale, that could for example result from a φ-dependent unified gauge coupling in some GUT theory [78]. The Yukawa couplings (3.34) mediate a long-range interaction by coherent scalarboson exchange between the fermions [60]. This interaction can be described by a Yukawa potential between two fermions of type i and j of spatial distance r U Yukawa (r) = g i g j e m φr r (3.35) with couplings 5 g i df i /dφ and the dynamic Quintessence mass m φ = V /2. As m φ is typically of the order H, inside the horizon (m φ r ) this interaction is a longrange interaction like gravity. Actually, it can be seen as a correction to the Newtonian potential: ( ) U(r) = G m i m j + Mpl 2 g i g j (3.36) r 4 L Yuk is a φ-dependent contribution to L B in the action (2.2). The mass, kinetic and φ-independent interaction terms of the fermions are also contained in L B. 5 To obtain this result one writes the φ field as its homogeneous part φ(t) plus a small fluctuation δφ and expands the Lagrangian in this small fluctuation. To first order in the fluctuation this gives a usual Yukawa term g i δφ ψψ with coupling g i = F i. The mass of the fluctuation δφ is given by the dynamical Quintessence mass m φ = V /2 [60, 78]. m i m j

38 EXTENDED QUINTESSENCE where the one in the brackets represents the Newtonian contribution and the second term the Quintessence contribution for an interaction of species i with j. One consequence of the species dependence is a violation of the equivalence principle. This turns out to put the most stringent bound on the couplings g i. The acceleration of different materials towards the sun has been shown to be the same up to one part in 0 0 from which a bound for the Yukawa couplings of neutrons and protons can be derived 6 [60]: g n, g p (3.37) This means a coupling of Quintessence to baryonic matter has to be highly suppressed. In other words, the strength of the interaction for baryonic matter is of the order g 2 n/m 2 n g 2 p/m 2 p (0 24 GeV) 2 and thus 0 orders of magnitude weaker than the gravitational coupling G (0 9 GeV) 2. In [6] it is pointed out that the long-range forces mediated by the Quintessence field could be suppressed by some symmetry. Another approach by Wetterich will be discussed in section (see [80, 8]). Note, however, that the couplings g i are, like the masses, φ-dependent quantities and thus they can also change with time. The constraints are only valid for today s values. Furthermore, a coupling between Quintessence and other forms of matter like leptons and dark matter are not restricted by the bound from the violation of the the equivalence principle. In the case of dark matter, bounds for a coupling to the Quintessence field arise e.g. from structure formation [2]. For Neutrinos, the coupling can be constrained by effects from the cosmic neutrino background on the Quintessence dynamics [39, 40] Varying Fundamental Constants Actually, not only the fermion masses (as discussed in the previous section) but basically all constants in the Standard model (and beyond) could depend on the Quintessence field 7. One often discussed possibility is a time variation of fundamental gauge couplings, that can be parameterized by L Gauge = 2 Z(φ)Tr(F µνf µν ) (3.38) where F µν is the field strength tensor of some gauge symmetry [82]. The time-dependent normalization can also be expressed by replacing the gauge coupling g according to g 2 g 2 /Z(φ) which leads to a time dependent effective coupling. For the photon field, this implies a time varying fine-structure constant α em. Actually, a detection of such a variation could be considered as a possible signal for Quintessence [24]. Furthermore, a variation in the strong coupling (and thereby the QCD scale) could lead to varying masses of baryons. If the standard model is embedded in a GUT theory, it is even possible to relate the variation of the various gauge couplings, yielding interrelations between the variation of nucleon masses and the fine-structure constant [82]. Thus Quintessence could predict a relation between the violation of the equivalence principle and the change of α em. 6 Numerically, this bound corresponds to M 2 plg 2 /m 2 < 0 0 where m is the nucleon mass. 7 The presence of the non-constant field φ will also alter the classical conservation laws since it is possible that e.g. energy and momentum is exchanged with the Quintessence field. However, the total energy and momentum are still conserved.

39 CHAPTER 3. QUINTESSENCE DYNAMICS 35 Actually, the effect of changing fundamental constants can show up in many different ways, giving the possibility to extract experimental bounds (see [82]). Besides geonuclear bounds (Oklo, α em (z 0.3) /α em < 0 7 ) and astronuclear bounds (decay rates in meteorites, α em (z 0.45) /α em < ), there are measurements from the observation of absorption lines in Quasars (typically α em (z 2)/α em with errors of the same order depending on which of the present experimantal results are believed [52, 69]). Furthermore, the question of changing constants and masses can also be addressed by nucleosynthesis and by CMB measurements [44,50,73] and, as describes above, tests of the equivalence principle. The typical feature of the Quintessence field to become important only recently could be related with the large changes of α em /α em over two orders of magnitude between z = 2 (if confirmed) and z < 0.3 [24, 82]. Altogether, the changes in the fundamental constants can only be very small during the evolution of the universe (e.g. BBN constrains Λ QCD (z 0 0 ) /Λ QCD < 0 2 and α em (z 0 0 ) /α em < 0 2 (0 4 ) where the latter bound applies if a GUTmotivated relation between α em and Λ QCD is used [44,73]) and therefore the functions Z(φ) (and F (φ) for the masses) may only vary slightly while φ changes of the order M pl or more during a Hubble time. As the Standard Model is supposed to be an effective field theory that emerges after integrating out some high energy degrees of freedom one could argue that one expects couplings that are suppressed by some high mass scale M, e.g. Z(φ) = βφ/m. Nevertheless, it turns out [6] that even for M of the order M pl it is typically required that β to fulfill the bounds and thus some symmetry is needed to further suppress such a coupling 8. The approach of Wetterich is to use dilatation symmetry and explain the small couplings by its radiative breaking (see section and [80, 8]) Non-Minimal Couplings Any action in curved space-time contains gravitational coupling terms since covariance requires a metric-dependent integrational measure g and the contraction of Lorentz indices also introduces a coupling to the metric (see action (2.2)). Couplings of the Quintessence field to geometrical quantities beyond these unavoidable couplings will be considered in this section. The geometrical quantities could e.g. be powers of the curvature scalar R or any combinations of the Ricci-tensor R µν and the curvature tensor R µνσρ that give singlets. In principle, the set could also be extended by considering couplings which contain space-time derivatives of the geometrical quantities or of the field φ or of both. It is well-known [3] that such couplings arise as quantum corrections in curved space-time, and this issue will be discussed in section 5.5. In general, non-minimal couplings are produced even when they are not present in the classical Lagrangian. Explicit curvature coupling will also modify the Einstein equations for gravity, like in scalar-tensor theories (introduced by Brans and Dicke [2]), leading for example to a time variation in Newton s constant [79]. Thus non-minimal couplings can be constrained by classical tests of gravity theories [9, 57]. To study the effects of non-minimal couplings quantitatively, one has to choose a special parameterization selecting only a small subset of all possible couplings. Taking 8 Without such a symmetry the naive expectation would be that β = O() [6].

40 EXTENDED QUINTESSENCE the curvature scalar (without derivative couplings), the most general contribution to the Lagrangian would be an arbitrary function of φ and R. The equations of motion in this case have been derived in [57]. However, it seems that without further assumptions one cannot make a statement about the viability of such models. A conventional choice is to restrict oneself to a Brans-Dicke like coupling linear in R but with an arbitrary function f(φ) [23, 57, 72] S = d 4 x ( g(x) R 6πG + f(φ)r + ) 2 ( φ)2 V (φ) + L B (3.39) leading to equations of motion ( R µν R ) 8πG 2 g µν [ ( ) = Tµν B + µ φ ν φ g µν 2 ( φ)2 V + 2f(φ)(R µν R/2g µν ) ] 2(f ;µ;ν g µν f), (3.40) φ + V (φ) Rf (φ) = 0. (3.4) Following the convention of Uzan [72] all φ-dependent quantities are written on the right hand side of the Einstein equation and the expression in the squared brackets is assigned to be the energy-momentum tensor T Q µν of the Quintessence field 9. With this prescription it is easy to check that the Friedman equations have the same form (2.9) as with f 0, with the only difference that the Quintessence energy density and pressure are now given by 20 ϱ φ = 2 φ 2 + V + 6H(Hf + f), (3.42a) p φ = 2 φ 2 V 2((2Ḣ + 3H2 )f + 2H f + f). (3.42b) For f = 0, one recovers the usual relations. One consequence of the non-minimal coupling is that the range of ω φ p φ /ϱ φ is not limited to [, +] any more 2 [7]. 9 Actually, there is no unique convention how to define the energy-momentum tensor (and thus densities and pressures) from the upper equations of motion. One can also shift the term proportional to the Einstein tensor R µν R/2g µν on the right hand side of (3.40) to the left hand side. The simplest possibility would then be to define the remaining terms on the right hand side of (3.40) as the energymomentum tensor. However, this definition leads to a violation of energy conservation [3] because the covariant divergence of this energy-momentum tensor is not equal to zero. One can also first shift the term proportional to the Einstein tensor on the right hand side of (3.40) to the left hand side and then divide the equation by /8πG 2f and absorb this factor into the definition of the Quintessence and the background energy-momentum tensor [57]. This convention leads to effective densities and pressures which conserve energy. However, in this case one runs into problems when f(φ) can get close to /6πG. For a detailed discussion see [3]. Note that the dynamics do not depend on the convention used, but the interpretation can change. 20 Note that the Quintessence energy-momentum tensor of Uzan has not the general form of the energy-momentum tensor of an ideal fluid (compare to (2.4)). For a homogeneous and isotropic universe (with Robertson-Walker metric) it is nevertheless possible to reduce the full Einstein equations to the Friedman equations by setting ϱ φ T Q 00 and p φ T ijδ ij /3a 2. For an ideal fluid, this definition coincides with the usual definition. 2 The usual argument that ω < allows superluminal velocities cannot be used here since the energy-momentum tensor has not the form of an ideal fluid. Note also that it is in principle possible that ϱ φ gets smaller than zero. Uzan [72] gives an example where this happens for a case where the coupling term dominates the whole action. Following Uzan, such solutions have to be excluded. Both features show up in all usual conventions for the choice of Tµν. Q

41 CHAPTER 3. QUINTESSENCE DYNAMICS 37 The modified equation of motion reads φ + 3H φ + V + 6(Ḣ + 2H2 )f = 0. (3.43) Within the upper framework, the only direct coupling term in the action which avoids the introduction of a new dimensionfull parameter is of the form φ 2 R. This corresponds to the choice f(φ) = 2 ξφ2 (3.44) with a dimensionless parameter ξ. Actually, this is the most commonly investigated extension [7, 9, 23, 3, 57, 72] and sometimes only this kind of coupling is denoted as non-minimal coupling. It has the special property that the Quintessence part of the action (3.39) is invariant under conformal transformations 22 if ξ = /6 ( conformal coupling ) and the potential V (φ) is either zero or proportional to φ 4. Stability of the tracking solutions Since non-minimal coupling terms are expected e.g. from Quantum corrections, it is important to study how sensitive the formalism of tracking solutions is to this corrections. Several studies have been made about this issue [9, 23, 72]. For the tracking solutions in exponential and inverse power law potentials, the typical order of magnitude of the Quintessence terms φ 2 as well as V (φ) in the action is Mpl 2 H2 for the exponential and φ 2 H 2 for the inverse power law potentials respectively. This means the non-minimal coupling f(φ)r is negligible if { M 2 f(φ) pl for exponential potentials φ 2 for inverse power law potentials (3.45) since R = (3ω B )H 2 /4. For a coupling of the form f = ξφ 2 /2 the upper inequality is clearly violated for the exponential potential, since φ is much larger than M pl for the tracking solution in the exponential potential. Indeed one finds that for ξ 0 tracking occurs only for the special case when the background is dominated by radiation (in this case the curvature scalar R vanishes) [72]. For the inverse power law potentials with a coupling f = ξφ 2 /2 one finds that the tracking attractor still exists and its stability is independent of ξ [9, 72]. The non-minimal coupling term has the same order of magnitude as the minimal terms for ξ = O() and will be suppressed for ξ. The non-minimal coupling does not modify the equation of state (3.4) of the tracking solution [9, 72]. However, another very important ingredient to a successful dark energy model is the late-time attractor in which dark energy dominates. Actually, we find that this attractor is profoundly modified by the non-minimal coupling f = ξφ 2 /2 in the inverse power law potential depending on the sign of ξ. For ξ > 0 the system approaches a Quintessence dominated de-sitter attractor with constant H and φ. φ α = H M α V (φ = ) pl 4 + α 3ξ 2ξ φ 2 (3.46) 22 A conformal transformation is a rescaling g µν s(x) 2 g µν, φ s(x) φ with an arbitrary spacetime dependent scalar function s(x).

42 EXTENDED QUINTESSENCE The universe inflates exponentially a e H t and one has (ωφ, Ω φ ) = (, ). This attractor is stable for all positive ξ. It can be understood as a balance between the forces V and Rf in the equation of motion. Since these terms only have opposite signs for ξ > 0, it is clear that the attractor can solely exist for positive ξ. For ξ < 0 the Quintessence field continues to roll down the potential after the cross-over, but with a different equation of state from the tracking solution. ω φ = + ξ 3 (α + 2)(α + 4) + ( ξ)(α 2) Ω φ = (3.47) Solutions with ξ < 3/(α 2 + 9α + 2) have to be excluded since they would lead to a negative energy density ϱ φ < 0. The behavior for both cases is shown on the left side of figure 3.8. For fixed inverse power law index α all solutions start at the same tracking fixed point (ω φ, Ω φ ) = ( + ( + ω B ) α α+2, ) but differ in their late time attractor as described above. For the transition between the two attractors, which is supposed to happen at the present epoch, figure 3.8 shows that ξ > 0 is more helpful to fulfill the present observational bound on the equation of state. However, there are more constraints in the case of non-minimal coupling which come from the modification of gravity (see next section). The region in parameter space where both the bounds from the modification of gravity (3.49, 3.50, 3.5) and the observational bound on the present equation of state (2.24) are fulfilled is shown on the right side of figure 3.8. One can see that ξ has to be constrained to rather small values ξ. The modification of gravity also has an impact on structure formation and the CMB which can give further constraints [57,73]. Ω φ Ω φ and ω φ today ξ= 0.09 ξ= 0.06 ξ= 0.03 ξ= 0 ξ= 0.02 ξ= ω φ Figure 3.8: Left: Phase plots in the (ω φ, Ω φ )-plane for an inverse power potential with α =.5 and various values of ξ. All trajectories start at the tracking fixed point on the x-axis. In the present epoch there is a transition to the de-sitter attractor (for ξ > 0) or an attractor with constant equation of state for ξ < 0 (see text). Compare to figure 3.5 for ξ = 0. Right: Allowed region in the parameter space (α, ξ). The red region is excluded from (3.50), the green region from the bound on ω φ (2.24), the purple from BBN (3.5) and the blue one from (3.49). The black region in the lower right corner has to be excluded since it allows ϱ φ < 0 (see text). Plots based on xpp.