West University of Timi¸soara Department of Physics 7 1 0 2 n a J 3 2 MASTER THESIS ] h t - p e h [ 1 v 3 4 2 6 0 . 1 0 SUPERVISOR, AUTHOR, 7 1 Lect.dr. NICOLAEVICI NISTOR AMBRUS¸ VICTOR EUGEN : v i X r a Timi¸soara 2010 West University of Timi¸soara Department of Physics PARTICLE PRODUCTION IN A ROBERTSON-WALKER SPACE WITH A DE SITTER PHASE OF FINITE EXTENSION SUPERVISOR, AUTHOR, Lect.dr. NICOLAEVICI NISTOR AMBRUS¸ VICTOR EUGEN Timi¸soara 2010 Abstract WeinvestigatethephenomenonofparticleproductioninaFriedmann-Robertson- WalkeruniversewhichcontainsaphaseofdeSitterexpansionforafiniteinterval, outside which it reduces to the flat Minkowski spacetime. We compute the par- ticle number density for a massive scalar and a spinorial field and point out differences between the two cases. We find that the resulting particle density ap- proaches a constant value at the scale of the Hubble time and that for a certain choice of the parameters the spectrum is precisely thermal for the spinorial field, and almost thermal for the scalar field. 3 4 Contents 1 Introduction 1 2 Quantum fields on curved spaces 5 2.1 Friedmann-Robertson-Walker spaces . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Quantization procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 The Klein-Gordon field . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2 The Dirac field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Particle production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1 The Klein-Gordon field . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.2 The Dirac field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 The free field equation on de Sitter space-time 23 3.1 The Klein-Gordon field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 The Dirac field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 The Minkowski solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4 Creation of massive scalar particles 33 4.1 Bogoliubov coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.1.1 de Sitter in and out modes . . . . . . . . . . . . . . . . . . . . . . . . 34 4.1.2 Mode mixing and density of created particles . . . . . . . . . . . . . . 36 4.1.3 Particle production of conformal massless scalar particles . . . . . . . 40 4.1.4 Graphical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Asymptotic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2.1 Large momentum p (cid:29) ω . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2.2 Low momentum q (cid:28) ω . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2.3 Middle region p (cid:28) ω (cid:28) q . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2.4 The number of created particles . . . . . . . . . . . . . . . . . . . . . 53 5 6 CONTENTS 4.2.5 Graphical comparison to the exact solution . . . . . . . . . . . . . . . 56 5 Creation of spinorial particles 61 5.1 Bogoliubov coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1.1 de Sitter in and out modes . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1.2 Mode mixing and density of created particles . . . . . . . . . . . . . . 63 5.1.3 Production of massless Dirac particles . . . . . . . . . . . . . . . . . 67 5.1.4 Graphical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Asymptotic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2.1 Large momentum p (cid:29) ω . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2.2 Low momentum q (cid:28) ω . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2.3 Middle region p (cid:28) ω (cid:28) q . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2.4 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2.5 The number of created particles . . . . . . . . . . . . . . . . . . . . . 79 5.2.6 Graphical comparison to the exact solution . . . . . . . . . . . . . . . 79 6 Conclusion 85 A Properties of Hankel functions 87 A.1 Differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 A.2 Asymptotic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 A.3 Hankel functions for the de Sitter scalar field . . . . . . . . . . . . . . . . . . 92 A.4 Hankel functions for the de Sitter spinorial field . . . . . . . . . . . . . . . . 92 B Pauli spinors 95 B.1 Spinor construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 B.2 Behaviour under parity transformations . . . . . . . . . . . . . . . . . . . . . 98 B.3 Relation between ξ and η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Chapter 1 Introduction The keywords describing this thesis are: quantum fields on curved spaces, de Sitter solutions, Bogoliubov transformation, particle production by the coupling with the gravitational field. Quantum fields on curved spaces are the generalization of the Minkowski quantum theory of fields. The general approach is to consider the space curvature as a background field, described by the metric tensor, which obeys the classical (i.e. not quantum) Einstein field equations. This approach suffers from a number of drawbacks, but it is nevertheless a bold step forward towards the grand unification of all known interactions and the quantization of the gravitational field. History tells us that a new theory is validated against results which are considered as classical. This pseudo-quantum treatment of quantum fields on a curved background provide a good way to produce some classical results. Working in a background gravitational field is all but easy. With very few exceptions, there are no known analytical solutions to the resulting field equations. One of these excep- tionsisthedeSitterspacetime, whichdescribesaFriedmann-Robertson-WalkerUniverseun- dergoing an exponential expansion (Misner, Thorne & Wheeler 1973, Birrell & Davies 1982). Even though solutions to field equations might be derived in an external gravitational field, the quantum theory built on them is not as natural and intuitive as it is on Minkowski. For example, the Poincar´e invariance of the Minkowski space and of the field equations automatically ensures the existence of “positive” and “negative” frequency states. However, a different choice of coordinate system (e.g. the spherical one), or the coordinates of a non- inertial observer (e.g. the Rindler coordinates) can also be used to define particle states, which might not have the same physical meaning as the former. The accelerated observer detects particles as if he would have been submerged in a thermal bath of temperature related to his own acceleration (Birrell & Davies 1982). We shall circumvent the more philosophical issues regarding particle states definition 1 2 CHAPTER 1. INTRODUCTION and interpretation, and instead consider the space to have only a finite region of a de Sitter expansion phase, outside which the space is flat (see Figure 1.1). a(cid:72)t(cid:76) Minkowski ddeeSSiitttteerr Minkowski t t t i f Figure 1.1: The scale factor for the FRW space under consideration Particle states are only defined on the flat regions of space. The method we employ is to lettheinvacuumstateevolvethroughthedeSitterphase,andcomputetheexpectationvalue of the particle number or the energy density operator in the out region. Poincar´e invariance guaranteesthereisnoparticleproductionontheMinkowskiregions. Thisphenomenontakes place only on the de Sitter phase. The particle and anti-particle modes used to construct the solution on de Sitter space are used merely as mathematical tools that allow us to propagate the in modes, and do not receive any physical interpretation as to their particle content. These key ingredients are summarized in Table 2.1. The purpose of this thesis is to evaluate the density of created particles with given momentum in a unit volume of the out space for a massive scalar and a spinorial field. We find that there is a cutoff for particles with momentum higher than the expansion factor. Using the spectral density we also evaluate the particle number density (per unit volume), and the energy density. We show that the particle number density approaches a constant value as the expansion time approaches the Hubble time, and it increases with the expansion factorandwiththemassofthecreatedparticles. Theenergydensityisfiniteonlyforthecase of a conformally coupled massive scalar field, in all other cases (including the spinorial field),