ABSTRACT QUANTUM COSMOLOGY: SOLUTIONS TO THE MODIFIED FRIEDMANN EQUATION In this work, a detailed analysis of Standard Cosmological Inflation is presented, which is then contrasted by Loop Quantum Cosmology (LQC), an application to cosmology from Loop Quantum Gravity (LQG). Specifically, the modified Friedmann equation of Loop Quantum Cosmology (LQC) is solved, in order to obtain expressions used to assess an Inflationary era in the early Universe. The expressions for the scale factor are derived when considering two regions associated with the behavior of the modified Friedmann equation, as well as the energy density and scalar field. The scale factor expression will then be used to provide a solution to the horizon problem that is related to the Big Bang model of the Universe, in contrast to what has been presented in the literature. James Anthony Rubio December 2016 QUANTUM COSMOLOGY: SOLUTIONS TO THE MODIFIED FRIEDMANN EQUATION by James Anthony Rubio A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Physics in the College of Science and Mathematics California State University, Fresno December 2016 APPROVED For the Department of Physics: We, the undersigned, certify that the thesis of the following student meets the required standards of scholarship, format, and style of the university and the student's graduate degree program for the awarding of the master's degree. James Anthony Rubio Thesis Author Gerardo Mu~noz(Chair) Physics Douglas Singleton Physics Frederick A. Ringwald Physics For the University Graduate Committee: Dean, Division of Graduate Studies AUTHORIZATION FOR REPRODUCTION OF MASTER'S THESIS I grant permission for the reproduction of this thesis in part or in its entirety without further authorization from me, on the condition that the person or agency requesting reproduction absorbs the cost and provides proper acknowledgment of authorship. X Permission to reproduce this thesis in part or in its entirety must be obtained from me. Signature of thesis author: ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Gerardo Mu~noz,for his continued guidance, wisdom, unwavering patience, and willingness to pursue many topics that the research directed us towards. Most importantly, I would like to thank him for his ability to help me understand difficult topics and ideas in physics. I would also like to thank my thesis committee members, Dr. Douglas Singleton and Dr. Frederick A. Ringwald, for all of their support and enlightening discussions during the writing process. Finally, I would like to thank the physics department faculty and staff for all of the knowledge attained in my time as a student and Teaching Associate. TABLE OF CONTENTS Page LIST OF FIGURES . vi INTRODUCTION . 1 The Big Bang . 3 STANDARD COSMOLOGICAL INFLATION . 7 Friedmann{Lema^ıtre–Robertson{Walker Universe . 7 Cosmic Inflation . 14 Issues of Cosmic Inflation . 26 Loop Quantum Cosmology . 27 SOLUTIONS TO THE MODIFIED FRIEDMANN EQUATION FROM LOOP QUANTUM COSMOLOGY . 30 Region I: t0 ≤ t ≤ tmax .......................... 32 Region II: t > tmax ............................ 39 ANALYSIS OF SOLUTIONS IN REGIONS I AND II . 43 Inflation in LQC . 43 The Horizon Problem . 45 CONCLUSION AND SUMMARY . 47 REFERENCES . 49 LIST OF FIGURES Page Figure 1. Typical “flat” potential, in relation to an inflaton scalar field. 22 Figure 2. Graphical representation of two regions of solutions corre- sponding to H. Region I corresponds to t0 ≤ t ≤ tmax and Region II corresponds to t > tmax. ..................... 32 To my family, for their continued support of my education throughout the years. INTRODUCTION Over many years, cosmology has seen several advances made as a study of the Universe. Astrophysical observations have been made for centuries and with time, they have increased knowledge of large objects in the sky, as well as the nature of the light itself. In terms of the Universe as a whole, a very significant observational study of galaxy redshifts was made in the early twentieth century [1], which led to a new realization and understanding among scientists. The results of the study made it clear that the majority of the galaxies were redshifted from our location in the Universe receding from our position. It was clear that the Universe was expanding with time. This expansion rate was measured at the time, and presently, the value is known within a small uncertainty. Another conclusion is that if the Universe is expanding with time, it had a finite, hot, and dense beginning. This early period of the Universe is referred to as the Big Bang. The Big Bang model of the Universe has had both observations on and off Earth that have strongly supported it as a scientific model of the Universe [2]. Measurements of light observed through astronomical telescopes have led to a strongly supported theoretical era of nucleosynthesis in the early history of the Universe, which serves as an explanation of the particle and chemical history of elements. Along with nucleosynthesis is an understanding of the thermal signature of the Cosmic Microwave Background observed in the sky, as observed photons scattered off of electrons. However, there are shortcomings of the Big Bang model. Representative shortcomings of the Big 2 Bang model are the flatness and horizon problems. The flatness problem refers to the fact that all observations point to a spatially flat Universe, while the Big Bang model allows for open, closed or flat solutions without a natural mechanism that would select the flat solution from either initial conditions or dynamical evolution. And the horizon problem arises from the observational data showing that the temperature of widely separated regions of the Universe is the same, despite the prediction of the Big Bang model that causal contact between these regions is not possible. The Cosmological Inflationary model is a large scope of this work, and provides a solution to both the flatness and horizon problems. Inflation precedes the Big Bang era, and is the subject of much research currently around the world. Many aspects of Inflation will be discussed as well as an alternative model that also provides a solution to the horizon problem. In this work, solutions are obtained from the modified Friedmann equation. The modified Friedmann equation is derived from Loop Quantum Cosmology, an application of a proposed quantum gravity model, by which space and time are quantized. Loop Quantum Cosmology (LQC), and the solutions obtained from it, provides a new formalism to evaluate the accelerated expansion of the Universe, and a reinterpretation of the horizon problem. The scope of this work is limited to flat LQC models. The general case will be addressed in future work. 3 The Big Bang As mentioned previously, the Big Bang model came about mostly as a result of astrophysical observations by both Vesto Slipher and Edwin Hubble [1, 3, 4]. Redshift, z; is defined as the ratio of the difference of an observed (λ0) and emitted (λ) wavelength of light, over the emitted wavelength, λ − λ z ≡ 0 (1) λ where if z > 0; then the galaxy is redshifted, and if z < 0; the galaxy is blueshifted. What was evident after observations by Hubble was that the majority of galaxies were redshifted, as opposed to blueshifted, especially for galaxies at high redshift. Hubble plotted the redshifts, z; versus the estimated distance r, from the galaxies and obtained a linear relationship, H z = 0 r (2) c where H0 is the Hubble Constant, currently measured at H0 = 73:24 ± 1:74 km s−1Mpc−1[5], where a Megaparsec, Mpc, is equal to 3:09 × 1022 m , and c is the speed of light, 3:00 × 108 m/s. The values measured by Hubble were small, so the assumption of non-relativistic Doppler shifts applied, which v meant that z = c . Therefore, the recessional speed of galaxies is proportional to the estimated distance, v = H0r : (3) 4 With (3), known as Hubble's Law, inferred from the observational data, the implication was that the Universe was expanding, and therefore has a finite age. The best estimate currently is (13:799 ± 0:021) × 109 years [6]. The concept of an expanding Universe with time had been considered theoretically as a consequence of General Relativity by Albert Einstein [7], the details of which will be covered in the next section. The idea of a once tiny, hot, and dense Universe paved the way for research into the chemical nature and composition of how the elements formed, specifically light elements. The idea behind nucleosynthesis is that the light elements, such as hydrogen, were formed when the Universe cooled from an initially hotter, denser radiation at temperatures that would not allow the binding energies of atom formation. In sum, as science has become knowledgeable of the energies associated with particles, predictions as to the elemental abundances present within the Universe support the notion of nucleosynthesis with current estimates from observation [8]. This is called the era of Big Bang Nucleosynthesis (BBN) and occurred when the Universe was in a hot, dense gaseous form [9]. Along with nucleosynthesis, strong support for the Big Bang model comes from the detection and properties of the Cosmic Microwave Background (CMB) of the Universe. In 1964, two researchers from Bell Telephone Laboratory, Arno A. Penzias and Robert W. Wilson, initially measured the CMB radiation which from thermodynamics, resembled blackbody radiation [7, 10]. The observation of a blackbody distribution immediately corresponded to a temperature that is currently measured at 2:72548 ± 0:00057 K [11]. This CMB temperature corresponds to photons scattered off of electrons when the Universe was about 5 380; 000 years old. Further observational data of the Cosmic Microwave Background (CMB) were measured by the COBE satellite in 1992, which showed an almost perfect blackbody spectrum of photons, with a temperature of T0 = 2:725 ± 0:001 K [12].