Viscous cosmology Bulk viscosity in the cosmological context Ben David Normann MSc in Physics Submission date: May 2015 Supervisor: Kåre Olaussen, IFY Co-supervisor: Iver Håkon Brevik, Institutt for energi- pg prosessteknikk (NTNU) Norwegian University of Science and Technology Department of Physics Bulk viscosity in cosmological context Mai 2015 MASTER’S THESIS Department of Natural Sciences, Institute of Physics Norwegian University of Science and Technology Main supervisor: Professor Kåre Olaussen External supervisor: Professor Iver H. Brevik i Preface The work contained in this book is a Master’s thesis in viscous cosmology, conducted at the physics department at the Norwegian University of Science and Technology (NTNU). The thesis was carried out within the international masters program of physics. The work was started the spring/summer of 2014 and ended mid mai 2015. The work has been carried out under the guidance of professor Iver Brevik at the department of Energy and Process Engineering at NTNU. Trondheim, May 15, 2015 Ben David Normann ii Acknowledgment First of all I would like to thank NTNU for providing me with the possibility of conducting a Master’s thesis. Thereafter I am above all thankful for the help and support from professor Iver Brevik, under whose supervision this work was conducted. His willingness to help, his kind attitude and his insight into the topic under investigation has been of great value. Also the many talks that went beyond physics have been source of much inspiration. I also want to thank my main supervisor, professor Kåre Olaussen, for his very useful input on a point where a push in the right direction was needed. I am no less thankful to professor Johan Skule Høye for his interest in the problem and his valuable contribution in establishing some of the estimates performed. I am also very grateful for the many educative talks and at times tempered discussions with both Marius Jakoby and John Ellingsen in the reading room during the process of writing. The mathematical input and creative mind-wandering with Rebecca Pretzsch was also both enjoyable and helpful - only surpassed by the extensive spell-check she so kindly helped me with. B.D.N. iii Summary and Conclusions The main objective of this thesis has been to estimate the present value of the bulk viscosity of the cosmic fluid. The following list contains a summary of the main results found in the present work: • The bulk viscosity is estimated for a two-component fluid consisting of pressure-less matter in mixture with the cosmological constant and also for a one-component fluid. Comparison with observations of the Hubble parameter as a function of redshift is performed. The investigation reveals that the magnitude of the viscosity is highly de- pendent on the model used. • The formalism and results found in a recent paper by J. Wang and X. Meng is discussed and put on vitally more solid ground through more accurate solutions of the energy equation together with the Friedmann equations. This is seen as one of the major contributions given through the present work. • For the one-fluid a degeneracy between the functional forms found in the literature for !(½) and ³(½) is pointed out. It is found instructive to use a constant equation of state parameter ! instead, and to include all inhomogeneity in ³(½). In this case the viscosity is bound to be negative with the form assumed for ³(½). • For a constant equation of state parameter in the one-fluid case, it is shown that - with the functional form considered for ³(½) - no other choice than ! 1 can explain Æ¡ observations well. • In general for the one-fluid case, ³ ½ is shown to give a rather good fit of the Hubble » parameter measurements, whereas ³ p½ by eye sight gives a clearly less correct fit. » • When allowing for a varying equation of state parameter !(½), however, both positive, negative and zero viscosity explains observations equally well. This is shown to be a consequence of having both a varying !(½) and ³(½). • An estimate obtained from small perturbations of the overall cosmic pressure is shown to suggest an order of magnitude ³ 106 Pa s for the present day viscosity. This is in 0 Æ good agreement with the amount of viscosity that is needed to make a visible change in the plotted curves of the Hubble parameter as a function of redshift. This is shown to be true for the two component fluid as well as for the one-fluid. • Finally, through applying kinetic theory, as well as using Winberg’s formalism, a non- negligible candidate is found as a cause for the viscosity; electromagnetic radiation slightly out of thermal equilibrium with matter. The present value of this viscosity is estimated to be ³ 103 Pa s. P Æ Contents Preface..............................................i Acknowledgment........................................ ii Summary and Conclusions.................................. iii 1 Introduction 2 1.1 Background........................................2 1.2 Objectives.........................................4 2 Establishing viscous cosmology5 2.1 Standard cosmology...................................5 2.1.1 Notation and Convention............................5 2.1.2 Mathematical toolbox..............................6 2.1.3 Einstein’s field equations............................8 2.1.4 The FRW-metric..................................9 2.1.5 Equation of state................................. 13 2.2 Short intermezzo on thermodynamics........................ 13 2.3 Incorporating dissipation................................ 15 2.3.1 Relativistic energy momentum tensor..................... 15 2.3.2 Expressions for the viscosity coefficients................... 19 2.3.3 Expanding Weinberg’s formalism....................... 22 2.3.4 Extending with general relativity........................ 24 2.4 Viscous cosmology.................................... 26 2.5 Observations........................................ 31 2.5.1 Hubble parameter................................ 31 2.5.2 Energy partitioning in the universe...................... 32 3 Classical kinetic theory 34 3.1 A closer look at dissipation and viscosity....................... 34 3.1.1 Kinetic theory................................... 34 3.1.2 The BGK-model and Chapman-Enskog solution.............. 45 3.1.3 Chapman-Enskog expansion of the Boltzmann equation......... 51 4 Constraining the bulk viscosity 52 4.1 Introduction........................................ 52 4.2 Literature survey..................................... 52 4.3 Article by Wang and Meng................................ 53 4.3.1 Derivation..................................... 54 4.3.2 Method....................................... 56 4.3.3 Results....................................... 56 4.4 General solutions for redshift-dependent viscosity................. 58 iv CONTENTS 1 4.4.1 Solving the energy equation........................... 58 4.4.2 Obtaining general expressions for E(z).................... 61 4.5 Solutions with constant bulk viscosity......................... 62 4.5.1 Matter domination in flat space........................ 62 4.6 General solutions with density dependent viscosity................. 64 4.6.1 Special case 1; constant viscosity........................ 65 4.6.2 Special case 2; viscosity proportional to H .................. 65 4.6.3 Special case 3; viscosity proportional to H 2 ................. 68 4.7 Solutions with temperature dependent viscosity................... 69 4.7.1 Obtaining ³(z)................................... 71 4.8 Effective one-fluid description found in literature.................. 71 4.8.1 Selecting ® and plotting............................. 72 4.8.2 Making sense of the one-component description.............. 77 4.8.3 Further tries with the one-fluid model.................... 78 4.9 Beyond phenomenology................................. 80 4.9.1 How big visc. can Zimdahl account for?.................... 81 4.9.2 How big viscosity can Weinberg account for?................. 83 4.9.3 The BGK formula................................. 85 4.9.4 Bulk viscosity as a first order modification of the pressure......... 87 5 Summary 89 5.1 Summary and conclusions............................... 89 5.2 Recommendations for further work.......................... 92 A Acronyms 94 B Christoffel Symbols for the FRW-metric 95 C Classical analogues for some of the tensors in the text 97 C.1 Classical decomposition................................. 97 C.2 Relativistic decomposition............................... 98 D Viscosity generation by perfect fluids 99 D.1 Functional form and interpretation.......................... 101 E Forth order Runge-Kutta method implemented in Matlab 102 E.1 Main Script (plotting and calling the numerical routine).............. 102 E.2 The function........................................ 105 F Script used for plotting solutions of Friedmann’s first Equation 107 F.1 Main script......................................... 107 F.2 The function........................................ 110 G Alternative ansatz for the energy equation 111 G.1 Interpretation....................................... 111 H Alternative justification of the BGK-formula 113 Bibliography 116 Chapter 1 Introduction 1.1 Background The so-called ¤CDM-model is often referred to as the standard cosmological model, pro- viding relatively simple explanations for many different observations, such as the acceler- ated expansion of the universe and the relative lack of anisotropy in the Cold Microwave Background radiation (CMB). The theory invokes solving Einstein’s field