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Hep-Th/0603057V3 16 Jun 2006 I.Osrainleiec O Akenergy Dark for Evidence Observational III

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Hep-Th/0603057V3 16 Jun 2006 I.Osrainleiec O Akenergy Dark for Evidence Observational III Dynamics of dark energy Edmund J. Copeland,1 M. Sami,2, 3 and Shinji Tsujikawa4 1School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom Email:[email protected] 2Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi, India 3Department of Physics, Jamia Millia Islamia, New Delhi, India Email:[email protected]; [email protected] 4Department of Physics, Gunma National College of Technology, Gunma 371-8530, Japan Email:[email protected] (Dated: February 1, 2008) In this paper we review in detail a number of approaches that have been adopted to try and explain the remarkable observation of our accelerating Universe. In particular we discuss the arguments for and recent progress made towards understanding the nature of dark energy. We review the observa- tional evidence for the current accelerated expansion of the universe and present a number of dark energy models in addition to the conventional cosmological constant, paying particular attention to scalar field models such as quintessence, K-essence, tachyon, phantom and dilatonic models. The importance of cosmological scaling solutions is emphasized when studying the dynamical system of scalar fields including coupled dark energy. We study the evolution of cosmological perturbations allowing us to confront them with the observation of the Cosmic Microwave Background and Large Scale Structure and demonstrate how it is possible in principle to reconstruct the equation of state of dark energy by also using Supernovae Ia observational data. We also discuss in detail the nature of tracking solutions in cosmology, particle physics and braneworld models of dark energy, the na- ture of possible future singularities, the effect of higher order curvature terms to avoid a Big Rip singularity, and approaches to modifying gravity which leads to a late-time accelerated expansion without recourse to a new form of dark energy. PACS numbers: 98.70.Vc Contents I. Introduction 3 II. Elements of FRW cosmology 5 A. Evolution equations 6 arXiv:hep-th/0603057v3 16 Jun 2006 B. The evolution of the universe filled with a perfect fluid 7 III. Observational evidence for dark energy 7 A. Luminosity distance 7 B. Constraints from Supernovae Ia 9 C. The age of the universe and the cosmological constant 10 D. Constraints from the CMB and LSS 12 IV. Cosmological constant 13 A. Introduction of Λ 13 B. Fine tuning problem 14 C. Λ from string theory 15 1. Four-form fluxes and quantization 15 2. The KKLT scenario 15 3. Relaxation of Λ in string theory 17 4. Λ from a self-tuning universe 17 5. Λ through mixing of degenerate vacua 17 D. Causal sets and Λ 18 E. Anthropic selection of Λ 18 2 F. A Dynamical Approach to the Cosmological Constant 19 G. Observing dark energy in the laboratory ? 19 V. Scalar-field models of dark energy 20 A. Quintessence 20 B. K-essence 22 C. Tachyon field 23 D. Phantom (ghost) field 24 E. Dilatonic dark energy 25 F. Chaplygin gas 26 VI. Cosmological dynamics of scalar fields in the presence of a barotropic perfect fluid 26 A. Autonomous system of scalar-field dark energy models 27 1. Fixed or critical points 27 2. Stability around the fixed points 27 B. Quintessence 28 1. Constant λ 28 2. Dynamically changing λ 30 C. Phantom fields 30 D. Tachyon fields 30 1. Constant λ 31 2. Dynamically changing λ 31 E. Dilatonic ghost condensate 33 VII. Scaling solutions in a general Cosmological background 34 A. General Lagrangian for the existence of scaling solution 34 B. General properties of scaling solutions 35 C. Effective potential corresponding to scaling solutions 36 1. Ordinary scalar fields 36 2. Tachyon 36 3. Dilatonic ghost condensate 36 D. Autonomous system in Einstein gravity 37 VIII. The details of quintessence 37 A. Nucleosynthesis constraint 37 B. Exit from a scaling regime 38 C. Assisted quintessence 38 D. Particle physics models of Quintessence 39 1. Supergravity inspired models 39 2. Pseudo-Nambu-Goldstone models 42 E. Quintessential inflation 43 IX. Coupled dark energy 44 A. Critical points for coupled Quintessence 45 B. Stability of critical points 45 1. Ordinary field (ǫ = +1) 46 2. Phantom field (ǫ = 1) 47 C. General properties of fixed− points 48 D. Can we have two scaling regimes ? 48 E. Varying mass neutrino scenario 50 F. Dark energy through brane-bulk energy exchange 50 X. Dark energy and varying alpha 51 A. Varying alpha from quintessence 51 B. Varying alpha from tachyon fields 52 XI. Perturbations in a universe with dark energy 54 A. Perturbation equations 54 B. Single-field system without a fluid 55 3 C. Evolution of matter perturbations 56 D. Perturbations in coupled dark energy 57 1. Analytic solutions in scalar-field matter dominant stage 57 2. Analytic solutions for scaling solutions 58 XII. Reconstruction of dark energy models 58 A. Application to specific cases 60 1. Case of p = f(X) V (φ) 60 2. Case of p = f(X)V−(φ) 60 3. Scaling solutions 60 B. Example of reconstruction 61 C. w = 1 crossing 61 − XIII. Observational constraints on the equation of state of dark energy 62 A. Parametrization of wDE 63 B. Observational constraints from SN Ia data 63 C. Observational constraints from CMB 65 D. Cross-correlation Tomography 68 E. Constraints from baryon oscillations 68 XIV. The fate of a dark energy universe–future singularities 69 A. Type I and III singularities 70 B. Type II singularity 70 C. Type IV singularity 70 XV. Dark energy with higher-order curvature corrections 71 A. Quantum effects from a conformal anomaly 71 B. String curvature corrections 72 XVI. Cosmic acceleration from modified gravity and other alternatives to dark energy 74 A. f(R) gravities 75 B. DGP model 77 C. Dark energy arising from the Trans-Planckian Regime 78 D. Acceleration due to the backreaction of cosmological perturbations 79 XVII. Conclusions 80 ACKNOWLEDGEMENTS 81 References 82 I. INTRODUCTION energy density during that period, leading to processes like inflation, baryogenesis, phase transitions etc... Now though we need to understand the impact particle physics Over the course of the past decade, evidence for has on cosmology today, how else can we explain the na- the most striking result in modern cosmology has been ture of this apparent cosmological constant? Theorists steadily growing, namely the existence of a cosmological never short of ideas, have come up with a number of constant which is driving the current acceleration of the particle physics related suggestions (as well as a num- Universe as first observed in Refs. [1, 2]. Although it may ber completely unrelated to particle physics) to help us not have come as such a surprise to a few theorists who understand the nature of the acceleration. were at that time considering the interplay between a number of different types of observations [3], for the ma- There is a key problem that we have to explain, and jority it came as something of a bombshell. The Universe it is fair to say it has yet to be understood. The value is not only expanding, it is accelerating. The results first of the energy density stored in the cosmological constant published in Refs. [1, 2] have caused a sea change in the today, which rather paradoxically is called dark energy way we have started thinking about the universe. and has nothing to do with dark matter, this value has 3 4 Conventionally, the world of particle physics and cos- to be of order the critical density, namely ρΛ 10− eV . mology has been seen as overlapping in the early uni- Unfortunately, no sensible explanation exists∼ as to why verse, particle physics providing much needed sources of a true cosmological constant should be at this scale, it 4 should naturally be much larger. Typically, since it is the dark energy equation of state, we have to probe back conventionally associated with the energy of the vacuum in time. A number of routes in that direction have been in quantum theory we expect it to have a size of order the suggested and plans are underway to extend this even typical scale of early Universe phase transitions. Even at further. For example by looking at the detailed patterns 3 4 the QCD scale it would imply a value ρΛ 10− GeV . of the anisotropies in the cosmic microwave background The question then remains, why has Λ got∼ the value it (CMB), we are seeing when and under what conditions has today? the photons left the surface of last scattering. As they Rather than dealing directly with the cosmological propagated towards us today, they will have traveled constant a number of alternative routes have been pro- through gravitational potentials determined by the na- posed which skirt around this thorny issue [4, 5, 6, 7, 8]. ture of the dark matter and dark energy, and so different They come in a a number of flavors. An incomplete forms of dark energy could in principle have led to differ- list includes: Quintessence models [9, 10] (see also ent contributions to quantities such as the separation of Refs. [11, 12]) which invoke an evolving canonical scalar CMB Peaks [48, 49, 50], the integrated Sachs Wolfe effect field with a potential (effectively providing an inflaton for [51], the nature of galaxy formation [52], the clustering of today) and makes use of the scaling properties [13, 14] large scale structure (LSS) as measured through quanti- and tracker nature [15, 16] of such scalar fields evolving ties such as σ8 [53, 54], the propagation of light through in the presence of other background matter fields; scalar weak and strong gravitational lenses [55, 56], and sim- field models where the small mass of the quintessence ply through the evolution of the Hubble expansion rate field is protected by an approximate global symmetry by itself which is a function of the energy contributions to making the field a pseudo-Nambu-Goldstone boson [17]; the Friedmann equation [57].
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