ABSTRACT Classi¯cation of Cosmological Models in Einstein's General Theory of Gravity Te Ha, M.S. Chairperson: Anzhong Wang, Ph.D. In this thesis, I ¯rst review the fundamentals of Hot Big Bang cosmology, the observational cosmology and the late accelerating universe. Then I systematically study the evolution of the Friedmann-Robertson-Walker (FRW) universe with a cosmological constant ¤ and a perfect fluid that has the equation of state p = w½, and classify all the solutions into various cases, where p and ½ are the pressure and energy density of the fluid, and w is a constant. In each case the main properties of the evolution of the universe are studied in detail, including the periods of deceleration or acceleration, and the existence of big bang, big crunch, and big rip singularities. Finally, I mention some future work along the direction laid down in this thesis. Classi¯cation of Cosmological Models in Einstein's General Theory of Gravity by Te Ha, B.S. A Thesis Approved by the Department of Physics Gregory A. Benesh, Ph.D., Chairperson Submitted to the Graduate Faculty of Baylor University in Partial Ful¯llment of the Requirements for the Degree of Master of Science Approved by the Thesis Committee Anzhong Wang, Ph.D., Chairperson Lorin Swint Matthews, Ph.D. Klaus Kirsten, Ph.D. Accepted by the Graduate School August 2009 J. Larry Lyon, Ph.D., Dean Page bearing signatures is kept on ¯le in the Graduate School. Copyright °c 2009 by Te Ha All rights reserved TABLE OF CONTENTS LIST OF FIGURES v ACKNOWLEDGMENTS vii DEDICATION viii 1 Hot Big Bang Cosmology 1 1.1 Relativistic Cosmology . 1 1.1.1 Cosmological Principle . 2 1.1.2 The Homogeneous and Isotropic FRW Universe . 5 1.1.3 Weyl's Postulate . 9 1.1.4 The Friedmann Equations . 10 1.2 Observational Cosmology . 12 1.2.1 Expanding Universe . 12 1.2.2 Big Bang Nucleosynthesis (BBN) . 13 1.2.3 Cosmic Microwave Background(CMB) . 14 1.3 The Late Cosmic Acceleration of the Universe . 16 2 Classi¯cation of Cosmological Models 18 2.1 Introduction to the FRW universe . 18 2.2 Classi¯cation of the FRW universe . 21 2.2.1 A. The k = 0 case . 23 2.2.2 B. The k = 1 case . 31 2.2.3 C. The k = ¡1 case . 40 iii 3 Summary and Future work 49 BIBLIOGRAPHY 51 iv LIST OF FIGURES 2.1 The potential given by Eq.(2.16) for k = 0 and ¤ > 0. 24 2.2 The expansion factor a(t), the accelerationa Ä(t), and the energy density ½(t) for k = 0 and ¤ > 0. The spacetime has a big bang singularity at t = 0 for w > ¡1. It is de Sitter for w = ¡1, which is free of any kind of spacetime singularities. When w < ¡1, a big rip singularity occurs at t = ts, at which we have a (ts) = 1 = ½ (ts). .......................... 26 2.3 The potential given by Eq.(2.16) for k = 0 and ¤ = 0. 28 2.4 The expansion factor a(t), the accelerationa Ä(t), and the energy density ½(t) for k = 0 and ¤ = 0. There is a big bang singularity at a = 0 for all the cases with w > ¡1. The spacetime is de Sitter for w = ¡1. When w < ¡1, a big rip singularity is developed at t = ts, at which we have a (ts) = ½ (ts) = 1. 28 2.5 The potential given by Eq.(2.29) for k = 0 and ¤ < 0. 30 2.6 The expansion factor a(t), the accelerationa Ä(t), and the energy density ½(t) for k = 0 and ¤ < 0. The spacetime is singular at a = 0 for all the cases with w > ¡1, (a big bang singularity). It is de Sitter for w = ¡1. When w < ¡1, a big rip singularity is developed at t = ts, at which we have a (ts) = ½ (ts) = 1. 31 1 2.7 The potential given by Eq.(2.38) for k = 1; w > ¡ 3 and ¤ > 0, where ¤c = ¤c (w; ½0). ................................ 33 2.8 The expansion factor a(t), the accelerationa Ä(t), and the energy density ½(t) 1 for k = 1; w > ¡ 3 and ¤ > 0, where ¤c is given by Eq.(2.40). A big bang singularity occurs at a = 0 in all cases with ¤ ¸ ¤c. In the ¯rst sub-case of 0 < ¤ < ¤c, both big bang and big crunch singularities occur, while in the second sub-case the spacetime is free of any kind of spacetime singularities. 34 1 2.9 The potential given by Eq.(2.42) for k = 1; w ·¡ 3 and ¤ > 0. 35 2.10 The expansion factor a(t), the accelerationa Ä(t), and the energy density ½(t) 1 for k = 1; w ·¡ 3 and ¤ > 0. A big bang singularity occurs only in the case w = ¡1=3 and C ¸ 1=2. In the case w < ¡1, a big rip singularity occurs at a = 1. ..................................... 36 2.11 The potential given by Eq.(2.44) for k = 1 and ¤ = 0. 37 v 2.12 The expansion factor a(t), the accelerationa Ä(t), and the energy density ½(t) for k = 1 and ¤ = 0. There are both big bang and big crunch singularities in the case w > ¡1=3, while only a big bang singularity occurs in the case w = ¡1=3. There is no singularity in the cases with ¡1 · w < ¡1=3. A big rip singularity occurs at a = 1 for w < ¡1. 38 2.13 The potential given by Eq.(2.45) for k = 1 and ¤ = 0: (a) for w > ¡1=3; (b) for w = ¡1=3 and C > 1=2; (c) for ¡1 < w < ¡1=3; (d) for w = ¡1 and C > j¤j=6; and (e) for w < ¡1. ........................ 39 2.14 The expansion factor a(t), the accelerationa Ä(t), and the energy density ½(t) for k = 1 and ¤ < 0. There are both big bang and big crunch singularities in the case with w > ¡1, while only a big bang singularity occurs in the case w = ¡1, and no singularities for the case w = ¡1, while there is a big rip singularity at a = 1 for w < ¡1. ....................... 39 2.15 The potential given by Eq.(2.46) for k = ¡1 and ¤ > 0. 41 2.16 The expansion factor a(t), the accelerationa Ä(t), and the energy density ½(t) for k = ¡1 and ¤ > 0. There are a big bang singularity for w > ¡1, no singularity for w = ¡1, and a big rip singularity at a = 1 for w < ¡1. 41 2.17 The potential given by Eq.(2.48) for k = ¡1 and ¤ = 0. 43 2.18 The expansion factor a(t), the accelerationa Ä(t), and the energy density ½(t) for k = ¡1 and ¤ = 0. There are a big bang singularity for w > ¡1, no singularity for w = ¡1, and a big rip singularity at a = 1 for w < ¡1. 43 2.19 The potential given by Eq.(2.51) for k = ¡1; ¤ < 0 and w > ¡1. 44 2.20 The expansion factor a(t), the accelerationa Ä(t), and the energy density ½(t) for k = ¡1; ¤ < 0 and w > ¡1. There are both big bang and big crunch singularities for all the cases with w > ¡1. ................... 45 2.21 The potential given by Eq.(2.51) for k = ¡1; ¤ < 0 and w = ¡1. 46 2.22 The expansion factor a(t), the accelerationa Ä(t), and the energy density ½(t) for k = ¡1; ¤ < 0 and w = ¡1. The spacetime is not singular in any of these cases. ...................................... 46 2.23 The potential given by Eq.(2.51) for k = ¡1; w < ¡1 and ¤ < 0, where ¤c is given by Eq.(2.52), where (a) is for j¤j > j¤cj; (b) is for j¤j = j¤cj; and (c) is for j¤j < j¤cj. ................................ 47 2.24 The expansion factor a(t), the accelerationa Ä(t), and the energy density ½(t) for k = ¡1; w < ¡1 and ¤ < 0, where ¤c is given by Eq.(2.52). There are big rip singularities in all the cases, except for the ¯rst sub-cases of j¤j > j¤cj and j¤j = j¤cj. ................................. 48 vi ACKNOWLEDGMENTS This thesis is written under the tremendous help of Dr. Anzhong Wang, Physics Department of Baylor University, who guided and directed my research for the whole year. I really appreciate his patience and dedication. In addition, I would also like to thank my fellow graduate students Yongqing Huang, Zhenbin Wu, Qianyu Ma, Kristen D. Pechan and Timothy J. Renner for the cooperation on the research about the classi¯cation of the FRW universe. I also thank the entire Physics department at Baylor University which o®ered me a very good environment to study and do the research. Special thanks to my parents, Qiushan Ha and Jun Yan, although they were not in the US, they gave me the most important support and love, which made me ¯nish the thesis. Finally, much thanks to the members of my thesis committee: Dr. Anzhong Wang, Dr. Lorin Swint Matthews and Dr. Klaus Kirsten. vii DEDICATION To my parents viii CHAPTER ONE Hot Big Bang Cosmology Hot Big Bang Cosmology is one hypothesis for the origin of the universe: At the very beginning, the universe was very small, the size almost less than a point nucleus, and was known as the singularity. However, it had great heat energy, and the heat overflowed the singularity from the extremely hot moment, so that ¯nally the explosion happened, which is called the Big Bang. This energy became the fundamental particles, and later these particles formed the substances, energy, space and time in the universe.