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The Ekpyrotic Universe

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The Ekpyrotic Universe The Ekpyrotic Universe Torquil MacDonald Sørensen Thesis submitted for the degree of Candidatus Scientiarum Department of Physics University of Oslo May 2005 Version 1.0, 2005-05-25 2 Acknowledgments I would like to thank my supervisor Prof. Øyvind Grøn for the opportunity to work in this exciting area of physics, and for valuable discussions and encouragement during my work. Also, Prof. Finn Ravndal has encouraged me to give lectures and seminars, for which I am very grateful. Thanks to all the students at the theory group. I especially mention my brother Atle, Eirik, Gerald, Ingunn, Mats, Morad, Nicolay, Olav and Tore. Most of all I would like to thank my family for their support. 3 4 Contents 1 Introduction 9 1.1 Space and Time . 9 1.2 Ekpyrotic Universe . 10 1.3 Summary of the thesis . 11 2 Strings and Branes 13 2.1 The bosonic string action . 13 2.2 Quantization . 24 2.3 Spectrum of physical states . 27 2.4 Unoriented strings . 30 2.5 Compactification . 32 2.6 Closed strings and one compact dimension . 34 2.7 T-Duality . 36 2.8 Several branes . 38 2.9 Unoriented strings and orientifolds . 39 2.10 D-brane action . 40 2.11 Super-symmetric strings . 41 2.12 M-theory . 42 3 Big Bang and Inflation 43 3.1 Introduction . 43 3.2 The Robertson-Walker metric . 43 3.3 Friedmann equations and the past singularity . 45 3.4 Cosmological problems . 48 3.5 Inflation . 50 3.6 Drawbacks of inflation . 52 4 Brane Universe 53 4.1 Introduction . 53 4.2 Motivation . 55 4.3 The induced Einstein equation . 56 4.4 Brane coupling to bulk gauge field . 60 5 5 Ekpyrotic Universe 63 5.1 The cosmological problems . 65 5.2 The ekpyrotic universe action . 68 5.3 Einstein equations . 72 5.4 Equations of motion for φ and . 77 A 5.5 Boundary conditions . 79 5.6 Time-independent solution . 80 5.7 The metric signature . 83 5.8 Geometric properties of the static solution . 84 5.9 Brane tensions . 86 6 Moduli Space Approximation and Numerical Results 87 6.1 Field theory of real, scalar fields . 87 6.2 The ekpyrotic universe . 89 6.3 Bulk brane potential . 96 6.4 Field equations for the ekpyrotic universe . 99 6.5 Numerical results . 102 6.6 Stability . 114 6.7 Dimensional bounce scenario . 116 6.8 Action with no bulk brane . 116 6.9 Brane matter . 118 7 Conclusion 121 A Moduli space approximation 123 B Einstein tensor 127 Bibliography 131 6 Units and conventions We use units with c = 1 and } = 1, and a metric signature of ( ; +; +; +), which • − gives positive square lengths for space-like vectors. The Ricci tensor is obtained from the Riemann tensor as follows: • α Rβδ := R βαδ: We define the “rationalized” Planck mass MD in D space-time dimensions by de- • manding that Einstein-Hilbert action is D 2 M − S := D dDxp g ; EH 2 − R Z where g is the determinant of the metric and is the Ricci curvature scalar. R The energy-momentum tensor is defined as a functional derivative of the action. It • is 2 δS T := m ; αβ −p g δgαβ − where Sm is the action for the gravitational source, e.g. matter, radiation and vacuum energy. The Einstein equation in D-dimensional space-time then become • 1 Eαβ = D 2 Tαβ; MD − where Eαβ is the Einstein tensor. The symmetric and antisymmetric part of a matrix Mαβ is denoted by parentheses • and square brackets, respectively, around the indices in question: 1 1 M := (Mαβ + Mβα) and M := (Mαβ Mβα) : (αβ) 2 [αβ] 2 − For any matrix we then have 1 M = M M : 2 (αβ) − [αβ] We define the wedge product between two one-forms to be • ! η := (! η η !) ; ^ ⊗ − ⊗ and similar for combinations of several one-forms. When an operator acts on a product of quantities, we use a parenthesis to indicate • the fact: @φ∂φ := (@φ) (@φ) = @ (φ∂φ). 6 7 8 Chapter 1 Introduction 1.1 Space and Time It is interesting to think about the evolution of our understanding of space-time, both in the accepted scientific theories and in the leading candidate for a universal descriptions of nature, string theory. Newtonian viewpoint The Newtonian viewpoint is that we have a universal time all over the universe, which is not affected by physical events. Time is simply a quantity used to parametrize the motion of objects. The difference in the coordinates of an event as seen from two inertial systems moving with constant speed with respect to each other is described by the Galilean transformations, under which the time coordinate is invariant. In the Newtonian world, gravitational influences between different objects propagate infinitely fast, since the gravitational field of an object depends on spatial distance, but not time. Einsteinian viewpoint Here time is an intrinsic, dynamical part of the geometry of the universe, and is not invariant under transformations between two inertial systems in relative motion. A clock that is moving with respect to the observer's reference system will run slower as seen by that observer. Another difference from the Newtonian viewpoint is that to preserve causality, no information can propagate faster that the speed of light, which is a constant velocity independent of the movements of the observer. Although possible also in the Newtonian world, here gravity is most often represented as curvature of space-time. The difference is that in the Einsteinian world the time coordinate is also affected by gravitational sources and the laws governing the four dimensional geometry (Einstein's field equations) are different from the corresponding three dimensional laws in a possible geometrical formulation of Newton's gravitation. Einstein gravity reduces to Newtonian gravity when only small masses and speeds are involved. Stringy viewpoint Although string theory can not be an accepted theory of nature since it has not been experimentally tested at the present time, it is certainly the best 9 CHAPTER 1. INTRODUCTION candidate we have for a theory of everything, by which we mean a theory which offers a quantum formulation of all the observed forces: electromagnetic force, weak nuclear force, strong nuclear force and the gravitational force. Nevertheless, since the ekpyrotic universe is based on string theory, we will assume string theory to be true throughout this thesis. That being said, string theory has an interesting new perspective on the geometry of space-time. Since string theory contains a quantum theory of gravitation, it predicts that space- time should be fuzzy at short distances. The fuzziness comes from the quantum corrections to the classical states, which are smooth. Another geometrical difference from theories with particles and curved space-time is how the laws of nature change when a dimension is compactified and shrunk to zero size. If we do this in standard quantum field theory, we get a theory with one less dimension. In string theory, due to the string's ability to wrap around the compactified dimension, we get back a theory with the same number of dimensions with which we started. A world in which one direction is compactified to the radius R can be interpreted as another world with a different string theory compactified with a radius of α0=R, where pα0 is the typical length of a string. This suggests that there is a minimal measurable length pα0. In string theory one has extended objects of various dimensions called D-branes (a generalization of the term membrane). In the realm of quantum mechanics, we know that observables corresponding to length in a certain direction, and momentum in the same direction don't commute. But we are accustomed to thinking that measurements of lengths in orthogonal directions commute. In string theory we can have a brane on which this isn't true, e.g. measuring the x-coordinate of an object living on the brane leads to an uncertainty when we try to measure its y-coordinate. This could be an indication that the underlying space-time is foamy instead of smooth. In general relativity the curvature of space-time is dynamic, but its topological prop- erties must be constant throughout the evolution of the geometry. String theory allows for changes in the space-time topology. Certain topological properties must stay fixed, however [1, 2, 3]. These topological transitions are by definition discontinuous, and can e.g. change the number of holes in space-time. 1.2 Ekpyrotic Universe Ever since it was discovered that super-symmetric string theory demands 10 space-time dimensions, it has been assumed by string theorists that at the low energies accessible now, six of those dimensions are curled up into a tiny compact manifold. In this picture our universe fills out all space, also the curled up dimensions. In the nineties it was discovered that the five existing string theories were different limits of a single theory, called M- theory. It is 11 dimensional and has 11 dimensional super-gravity as its low energy limit. The fundamental objects it contains are not zero dimensional as the particles in quantum field theory, but M2- and M5-branes (having 2 and 5 spatial dimensions), and the objects in the five familiar 10 dimensional string theories (strings and branes) are in fact obtained 10 1.3. SUMMARY OF THE THESIS by wrapping these M2- and M5-branes around a small compact dimension. The non- perturbative part of M-theory is mostly unknown at the present time, but its low energy limit is known as 11 dimensional super-gravity, and that theory has been studied since the 1970s.
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