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The Cosmological Constant Problem, an Inspiration for New Physics

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The Cosmological Constant Problem, an Inspiration for New Physics ITP-UU-06/34 SPIN-06/30 gr-qc/yyxxxxx The Cosmological Constant Problem, an Inspiration for New Physics Stefan Nobbenhuis Ph.D. Thesis, defended June 15. 2006 Supervisor: Prof. Dr. G. ’t Hooft. Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands and Spinoza Institute Postbox 80.195 3508 TD Utrecht, the Netherlands [email protected] arXiv:gr-qc/0609011v1 4 Sep 2006 We have critically compared different approaches to the cosmological constant problem, which is at the edge of elementary particle physics and cosmology. This problem is deeply connected with the difficulties formulating a theory of quantum gravity. After the 1998 discovery that our universe’s expansion is accelerating, the cosmological constant problem has obtained a new dimension. We are mainly interested in the question why the cosmological constant is so small. We have identified four different classes of solutions: a symmetry, a back-reaction mechanism, a violation of (some of) the building blocks of general relativity, and statistical approaches. In this thesis we carefully study all known potential candidates for a solution, but conclude that so far none of the approaches gives a satisfactory solution. A symmetry would be the most elegant solution and we study a new symmetry under transformation to imaginary spacetime. Contents 1 The Cosmological Constant Problem 1 1.1 The ‘Old’ Problem, Vacuum Energy as Observable Effect . ........ 1 1.1.1 Reality of Zero-Point Energies . ... 5 1.1.2 RepulsiveGravity ............................. 5 1.2 TwoAdditionalProblems . 6 1.3 Renormalization ................................. 7 1.4 Interpretations of a Cosmological Constant . ......... 10 1.4.1 Cosmological Constant as Lagrange Multiplier . ....... 10 1.4.2 Cosmological Constant as Constant of Integration . ........ 10 1.5 WheretoLookforaSolution? . 11 1.5.1 Weinberg’s No-Go Theorem . 13 1.5.2 SomeOptimisticNumerology . 15 1.6 Outlineofthisthesis ............................. 16 2 Cosmological Consequences of Non-Zero Λ 19 2.1 TheExpandingUniverse. 19 2.1.1 SomeHistoricObjections . 22 2.2 Some Characteristics of FRW Models . ..... 23 2.2.1 Redshift................................... 25 2.3 TheEarlyUniverse................................ 28 2.3.1 Deriving Geometric Information from CMBR Anisotropies....... 31 2.3.2 EnergyDensityintheUniverse . 34 2.4 EvolutionoftheUniverse . 36 2.5 Precision cosmology in recent and upcoming experiments ........... 38 2.5.1 SupernovaeTypeIa ............................ 38 2.5.2 PioneerAnomaly.............................. 40 3 4 CONTENTS 3 Type I: Symmetry Mechanism 43 3.1 Supersymmetry................................... 44 3.1.1 No-ScaleSUGRA.............................. 45 3.1.2 UnbrokenSUSY .............................. 47 3.1.3 Non-SUSYStringModels .. .. .. .. .. .. .. 48 3.2 Scale Invariance, e.g. Conformal Symmetry . ........ 48 3.2.1 Λ as Integration Constant . 50 3.3 Holography ..................................... 51 3.3.1 Conceptual Issues in de Sitter Space . 53 3.4 “Symmetry” between Sub- and Super-Planckian Degrees of Freedom . 54 3.5 Interacting Universes, Antipodal Symmetry . ......... 56 3.6 DualityTransformations . 57 3.6.1 S-Duality .................................. 57 3.6.2 HodgeDuality ............................... 58 3.7 Summary ...................................... 58 4 Invariance Under Complex Transformations 59 4.1 Introduction.................................... 59 4.2 ClassicalScalarField. 60 4.3 Gravity ....................................... 61 4.4 Non-relativistic Particle . ...... 62 4.5 HarmonicOscillator .............................. 64 4.6 SecondQuantization .............................. 65 4.7 PureMaxwellFields ............................... 69 4.8 Relation with Boundary Conditions . ..... 70 4.9 RelatedSymmetries ............................... 72 4.9.1 Energy Energy ............................ 74 →− 4.10Summary ...................................... 75 5 Type II: Back-reaction 77 5.1 Scalar Field, Instabilities in dS-Space ...................... 77 5.1.1 ‘Cosmon’Screening ............................ 80 5.1.2 Radiative Stability in Scalar Field Feedback Mechanism........ 82 5.1.3 Dilaton ................................... 82 5.2 Instabilities of dS-Space.............................. 83 CONTENTS 5 5.2.1 Scalar-type Perturbations . 83 5.3 RunningΛfromRenormalizationGroup . ..... 87 5.3.1 Triviality as in λφ4 Theory ........................ 92 5.4 Screening as a Consequence of the Trace Anomaly . ....... 93 5.5 Summary ...................................... 94 6 Do Infrared Gravitons Screen the Cosmological Constant? 95 6.1 Introduction.................................... 95 6.2 AReviewoftheScenario ............................ 96 6.2.1 ANewtonianPicture............................ 98 6.3 Evaluation...................................... 99 6.4 The Full Quantum Gravitational Calculation . ........ 100 6.5 Renormalization in Perturbative QG . 105 6.6 Summary ...................................... 106 7 Type III: Violating the Equivalence Principle 109 7.1 Extra Dimensions, Braneworld Models . 109 7.2 Randall-Sundrum Models, Warped Extra Dimensions . ......... 110 7.2.1 Self-TuningSolutions . 112 7.2.2 ExtraTime-likeDimensions . 114 7.3 InfiniteVolumeExtraDimensions . 114 7.3.1 MassiveGravitons ............................. 121 7.3.2 Non-ZeroBraneWidth .......................... 124 7.3.3 Non-localGravity ............................. 124 7.3.4 A 5D DGP Brane-World Model . 126 7.4 Ghost Condensation or Gravitational Higgs Mechanism . .......... 126 7.5 FatGravitons.................................... 129 7.6 Composite Graviton as Goldstone boson . 131 7.6.1 Summary .................................. 132 8 Type IV: Statistical Approaches 135 8.1 HawkingStatistics ............................... 137 8.1.1 Wormholes ................................. 139 8.2 AnthropicPrinciple. .. .. .. .. .. .. .. .. 142 8.2.1 Anthropic Prediction for the Cosmological Constant . ......... 143 6 CONTENTS 8.2.2 Discrete vs. Continuous Anthropic Principle . ........ 146 8.3 Summary ...................................... 146 9 Conclusions 149 9.1 FutureEvolutionoftheUniverse . 150 A Conventions and Definitions 151 B Measured Values of Different Cosmological Parameters 153 B.1 HubbleConstant .................................. 153 B.2 TotalEnergyDensity.. .. .. .. .. .. .. .. .. 154 B.3 MatterintheUniverse. .. .. .. .. .. .. .. .. 154 B.4 DarkEnergyEquationofState . 154 B.5 Summary ...................................... 155 Bibliography 177 Chapter 1 The Cosmological Constant Problem In this chapter we carefully discuss the different aspects of the cosmological constant problem. How it occurs, what the assumptions are, and where the difficulties lie in renormalizing vacuum energy. Different interpretations of the cosmological constant have been proposed and we briefly discuss these. We subsequently, as a warm-up, point at some directions to look at for a solution, before ending this chapter by giving an outline of the rest of the thesis. 1.1 The ‘Old’ Problem, Vacuum Energy as Observable Effect 1 In quantum mechanics, the energy spectrum, En, of a simple harmonic oscillator is given by En = (n+1/2)ω. The energy of the ground state, the state with lowest possible energy, is non- zero, contrary to classical mechanics. This so-called zero-point energy is usually interpreted by referring to the uncertainty principle: a particle can never completely come to a halt. Free quantum field theory is formulated as an infinite series of simple harmonic oscillators. The energy density of the vacuum in for example free scalar field theory therefore receives an infinite positive contribution from the zero-point energies of the various modes of oscillation: 3 ∞ d k 1 2 2 ρ = 3 k + m , (1.1) h i 0 (2π) 2 Z p 2 2 and we set ω(k)= √k + m . With a UV-cutoff kmax, regularizing the a priori infinite value 2 4 for the vacuum energy density, this integral diverges as kmax. The filling of the ‘Dirac sea’ in the quantization of the free fermion theory leads to a downward shift in the vacuum energy with a similar ultraviolet divergence. The Hamiltonian is: 3 d k s s s s H = ω a †a b †b , (1.2) (2π)3 − s Z X 1See Appendix A for conventions and some definitions used throughout this thesis, e.g. ~ = 1. 2This is just the leading divergence, see section (1.3) for a more precise discussion. 1 2 CHAPTER 1. THE COSMOLOGICAL CONSTANT PROBLEM s where the bk† creates negative energy. Therefore we have to require that these operators satisfy anti-commutation relations: r s 3 (3) rs b , b †′ = (2π) δ (k k′)δ , (1.3) { k k } − r r since this is symmetric between b and b †, we can redefine the operators as follows: s s s s ˜b b †; ˜b † b , (1.4) ≡ ≡ which obey the same anti-commutation relations. Now the second term in the Hamiltonian becomes: kmax 3 s s s s d k ωb †b =+ω˜b †˜b (const.); (const) = ω(k). (1.5) − − (2π)3 Z Now we choose 0 to be the vacuum state that is annihilated by the as and ˜bs and all | i excitations have positive energy. All the negative energy states are filled; this is the Dirac sea. A hole in the sea corresponds to an excitation of the ground state with positive energy, compared to the ground state. The infinite constant contribution to the vacuum energy density has the same form as in the bosonic case, but enters with opposite sign. Spontaneous symmetry breaking gives a finite but still possibly large shift in the vacuum energy density. In this case, 1 = gµν ∂ φ∂ φ V (φ) (1.6) L 2 µ ν − where the potential
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