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Dynamical Approach to the Cosmological Constant Problem

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Dynamical Approach to the Cosmological Constant Problem Dynamical approach to the cosmological constant problem Zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN von der FakultÄatfÄurPhysik des Karlsruher Instituts fÄurTechnologie (KIT) genehmigte DISSERTATION von MS Viacheslav A. Emelyanov aus Moskau (Russische FÄoderation) Karlsruher Institut fÄurTechnologie { Campus SÄud FakultÄatfÄurPhysik Institut fÄurTheoretische Physik Tag der mÄundlichen PrÄufung:2. November 2012 Referent: Prof. Dr. F. R. Klinkhamer (Institut fÄurTheoretische Physik, KIT) Korreferent: Prof. Dr. H. Nicolai (Max-Planck-Institut fÄurGravitationsphysik, AEI) Contents Conventions and Abbreviations 3 1 Introduction 4 2 Theoretical framework 6 2.1 Einstein-Hilbert action . 6 2.2 Robertson-Walker metric . 7 2.3 Matter composition of the universe . 7 2.4 Cosmological evolution of the universe . 8 2.5 Newton's law of gravity . 9 3 Cosmological constant problem 11 3.1 Zero-point energy of quantum ¯elds . 11 3.2 Semi-classical Einstein equations . 13 3.3 Cosmological constant problem . 14 3.3.1 Cosmological constant and zero-point energy . 14 3.3.2 Cosmological constant and spontaneous symmetry breaking . 17 4 Particular approaches to CCP 21 4.1 Fine-tuning adjustment . 21 4.2 Dynamical adjustment . 22 4.3 Q-theory . 23 5 Vector-tensor model I 26 5.1 Vector and Einstein ¯eld equations . 27 5.2 Flat, homogeneous and isotropic universe . 28 5.2.1 Case: A¹ = (A0; 0) . 28 5.2.2 Case: A¹ = (0;Ai) ......................... 38 5.2.3 Case: A¹ = (A0;Ai) ........................ 42 5.3 General linear perturbations and Newton's law of gravity . 45 1 6 Vector-tensor model II 48 6.1 Vector and Einstein ¯eld equations . 49 6.2 Flat, homogeneous and isotropic universe . 50 6.3 General linear perturbations and Newton's law of gravity . 55 7 Discussion 58 8 Concluding remarks 60 Appendices 62 Appendix A Backreaction of quantum scalar ¯eld on metric . 62 Appendix B One-loop vacuum energy . 65 Appendix C Vector energy-momentum tensor . 67 Appendix D General linear perturbation of vector ¯eld equation and energy- momentum tensor . 69 Appendix E Energy-momentum tensor of vector ¯elds . 73 Appendix F Expansion coe±cients . 74 Appendix G General linear perturbation of vector ¯eld equations and energy- momentum tensor of vector ¯elds . 78 Acknowledgements 82 References 83 2 Conventions and Abbreviations We work throughout in units implying ~ = c = 1, where ~ is the reduced Planck constant and c is the speed of light in vacuum. In what follows, we consider that spacetime is a four-dimensional, globally hyperbolic manifold M with a metric g. We specify every point on M as x ´ x¹ = (x0; x1; x2; x3), where x0 is time and xi, where i = 1; 2; 3 are spatial coordinates. Greek indices always run over 0; 1; 2; 3, while roman ones denote only spatial components. ¹ º We designate the metric components by g¹º, i.e. g = g¹º dx ­dx , with the signature (+; ¡; ¡; ¡). At certain points in this work, we will use two special con¯gurations of g¹º, namely ±¹º and ´¹º. The former represents the metric of the four-dimensional 4 4 Euclidean space E , and the latter is the metric of Minkowski spacetime M1;3. The a±ne connection r is assumed to be the Levi-Civita connection. Thus the ¸ connection coe±cients ¡ ¹º are given by the Christo®el symbols: 1 ³ ´ ¡¸ = g¸½ @ g + @ g ¡ @ g ; ¹º 2 ¹ º½ º ¹½ ½ ¹º ¹ ¹ where @¹ is the partial derivative with respect to x , i.e. @=@x . We will frequently use shorthand notations (:::); ¹ and (:::);¹ for denoting the covariant and partial derivatives, respectively. ¹ The components of the Riemannian curvature tensor R º¸½ are given by ¹ ¹ ¹ ¹ σ ¹ σ R º¸½ ´ @¸¡ º½ ¡ @½¡ º¸ + ¡ σ¸¡ º½ ¡ ¡ 嫭 º¸ : ¸ ¹º The Ricci tensor and scalar are de¯ned as R¹º ´ R ¹¸º and R ´ g R¹º, respectively. 3 Chapter 1 Introduction Recent observations of type Ia supernovae have shown that the expansion of the universe is accelerating [1]. This phase has started at redshift z » 1, that corresponds to the epoch of galaxy formation. These facts were later con¯rmed by other experiments [2, 3]. This phenomenon goes under the name \dark energy". The outstanding question is to discover the physics of it. The up-to-date experimental data show that its energy density ½de is approximately equal to 72:8% of the critical mass density: ¡47 4 ½de ¼ 2:9 £ 10 (GeV) ; the pressure Pde of it is negative, with the absolute value roughly equaling its energy density, i.e. Pde ¼ ¡½de. Many proposals have been suggested in an attempt to uncover its nature, among which is the cosmological constant associated with the lowest-energy state of quantum ¯elds. The fact is that quantum theory implies that the normal or ground state of a certain ¯eld is endowed with the zero-point or vacuum energy density ½V, which vanishes in the classical limit ~ ! 0. Vacuum also possesses the pressure PV, that is related to its energy density as PV = ¡½V. Naive theoretical estimates give an unacceptably large value of the zero-point energy density ½V, that strongly contradicts the observations. In addition, the nonzero Higgs condensate in the standard electroweak theory and the quark and gluon condensates in quantum chromodynamics make enormous contributions to the total vacuum energy as well. This is the essence of the cosmological constant problem (CCP). In addition to this puzzle, there are two related cosmological problems. Speci¯cally, it is asked why ½de is not precisely zero, but of the order of the matter energy density ½m of the universe, taking into account that ½de and ½m depend di®erently on cosmic time. The latter is the so-called cosmic coincidence problem. In the present research, we address the question of how to dynamically get rid of the 4 large value of the total vacuum energy produced by quantum ¯elds. This thesis is arranged as follows: in Chapter 2, we will briefly describe our the- oretical framework that will be used throughout this work. Then, in Chapter 3, we will talk over the ¯rst cosmological constant problem in detail. In Chapter 4, particu- lar approaches to the problem are briefly discussed. These are ¯ne-tuning, dynamical adjustment and q-theory [4]. Motivated by q-theory and its special realization [5, 6], we will treat in Chapter 5 vector-tensor model giving a dynamical cancellation of the total vacuum energy appearing in the Einstein equations. We will also discuss a serious obstacle inherent in it. In Chapter 6, we will considerably modify this model in order to overcome that flaw. And, ¯nally, in Chapters 7 and 8, we will discuss our results and conclude. 5 Chapter 2 Theoretical framework According to contemporary physics, there exist four fundamental interactions in nature. These are the electromagnetic, weak, strong and gravitational forces. The ¯rst three interactions are well-described by gauge vector bosons according to the standard model of particle physics based on quantum ¯eld theory with the local symmetry group U(1)Y £ SU(2)L £ SU(3)C [7, 8]. General relativity (GR) is a classical theory of gravity. This theory is invariant under Di®(M) group and based on geometrical and dynamical (equivalence) principles [9]. The geometrical idea implies spacetime is a torsion-free manifold that locally looks as 4 M1;3. Thus the metric g contains all information about the gravity. The equivalence principle means that inertial mass coincides with the gravitational one. The weak and strong interactions are short-range, while the electromagnetism and gravity are long-range. However, matter is electrically neutral on average, so that gravity governs the evolution of the universe on large scales. 2.1 Einstein-Hilbert action The dynamics of the metric in general relativity is determined by Z 1 p S [g] = ¡ d4x ¡g R ; (2.1) EH 16¼G where the integration is performed over the manifold M with @M = 0. This action was proposed by Hilbert in 1915 and is known as the Einstein-Hilbert action functional, where G is the gravitational constant [9]. The Einstein ¯eld equations are derived by setting the functional derivative of (2.1) over g¹º to zero. Thus one has 1 G ´ R ¡ R g = 8¼G T m ; (2.2) ¹º ¹º 2 ¹º ¹º 6 where we have also added a matter action Sm[g; Ã] to (2.1) with an energy-momentum tensor (EMT) de¯ned in the usual manner: 2 ±Sm[g; Ã] T m (x) ´ p ; (2.3) ¹º ¡g ±g¹º(x) and G¹º in (2.2) is the Einstein tensor. 2.2 Robertson-Walker metric Observational data show that the universe is homogeneous and isotropic on scales larger than 100 Mpc. The metric tensor g of such the universe is µ ¶ dr ­ dr ³ ´ dt ­ dt ¡ a2(t) + r2 dθ ­ dθ + sin2 θ d' ­ d' (2.4) 1 ¡ kr2 and known as the Robertson-Walker metric, where a(t) is a scale factor and the curva- ture constant k 2 f¡1; 0; +1g. These values of k correspond to open, flat and closed universes, respectively. Observations indicate that our universe is roughly flat, therefore we set k = 0 in what follows. Having used (2.4) with k = 0, one ¯nds the nonzero components of G¹º: ¡ ¢ 2 _ 2 G00 = 3H and Gij = 2H + 3H gij ; (2.5) where dot stands for a di®erentiation over t and H ´ a=a_ is the Hubble parameter. m m Note, the symmetries of M encoded in (2.4) imply T0i = 0 and Tij / gij as these directly follow from (2.2) and (2.5). 2.3 Matter composition of the universe Cosmic microwave background radiation The cosmic microwave background (CMB) comprises primordial photons that decoupled from matter after the recombination epoch zrec ¼ 1100.
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