Cosmological Models with Quintessence: Dynamical Properties and Observational Constraints Shao-Chin Cindy Ng Department of Physics and Mathematical Physics Adelaide lJniversity South Australia, Australia 1-1- September 2OOL Contents Abstract llI Declaration iv Acknowledgements v 1 Introduction 1 1.1 Quintessence as Missing Energy 1 L2 Cosmological Models with Quintessence. 4 Lz1 Inverse Power-Law Potential 4 L.2.2 Simple Exponential Potential . 6 r.2.3 Double-Exponential Potential I r.2.4 Pseudo Nambu-Goldstone Bosons (PNGB) . 7 1.3 Quintessential Difficulties . 9 1.4 Outlines of Chapters 9 2 Phase-Space Analysis 11 2.7 Simple Exponential Potential . 12 2.2 Inverse Power-Law Potential 13 2.2.1 Kinetic Energy Domination and Frozen Field 15 2.2.2 Tracker Solution 17 2.2.3 Inflationary Solution 18 2.3 Doubie-Exponential Potential 19 2.4 PNGB Potential 22 2.4.1 Field Variables 23 2.4.2 Energy Density Variables 28 3 Numericallntegration 32 3.1 Simple Exponential Potential 32 3.2 Inverse Power Law Potential , 35 .tù 3.3 Double-Exponential Potential .)f 3.4 PNGB Potential 39 3.5 Luminosity Distance 44 4 Supernovae Tests 46 4.I Constraints on Quintessential Parameters . 47 4.2 Evolution of Supernovae 54 4.3 Model Compartson bb 4.3.I Accelerating Universe versus Luminosity Evolution 6( 4.3.2 PNGB versus Other Potentials 70 5 Future Supernovae Probes 73 5.1 SNAP Data Simulation 74 5.2 Likelihood Function 76 5.3 Model Compartson 77 5.3.1 Accelerating Universe versus Luminosity Evolution 77 5.3.2 PNGB versus Other Potentials 80 5.4 Discussion 81 6 Gravitational Lensing Statistics Test 84 6.1 Likelihood Function 85 6.2 Constraints on Quintessence Models . 87 7 Conclusion 94 7.7 Discussion 94 7.2 Further Studies 97 Bibliography 100 Publications LO7 11 Abstract Many recent astronomical observations indicate that ordinary gravitationally clumped matter constitutes of order' 113 of the total energy density of the universe, and that the remaining 213 should take the folm of a smooth component which does not clump significantly gravitationally and which is quite unlike any form of rnatter observed in the laboratory. One of the candidates for the "missing energy" is the self-interacting energy of a dynarnical scalar field with its potential, which is given the name "quintessence". This thesis aims to study different quintessence models, in particular, a quintessence arising from an ultra-light pseudo Nambu- Goldstone boson (PNGB). The dynamical properties of the different quintessence models are overvìewed by phase-space analyses, to study the attractor and "tracket" solutions in the phase space; the existence of these solutions could avoid fine tuning of the initial conditions. The high-redshift type Ia supernovae constraints upon these quintessence models are studied. With the PNGB models, the impact of existence of a simple phenomenological model for supernovae luminosity evolution to the supernovae constraints is also studied. The potentials of a future supernovae data set, as might be obtained by the proposed SuperNova Acceleration Probe (SNAP) satellite, to discriminate the PNGB models over the other quintessence models and to discriminate the existence of the supernovae luminosity evolution are studied. Finally, the gravitional lensing statistics of high luminosity quasars constraints upon the quintessence models are briefly discussed' llr Acknowledgements I wish to thank the Adelaide University for providing me with an International Postgraduate Research Scholarship (IPRS), and a Research Abroad Scholarship to partiy support my study leave in United Kingdom. I wish to thank my supervisor, Dr. David Wiltshire, for inspiring my interest into this area of research, and for four years of guidance. I also wish to thank David for sparing a moment out of his busy schedule to see me everytime I needed an advice at work. I wish to thank the staff of the Centre for Theoretical Physics at University of Sussex, Unitecl Kingdom, for their kind hospitality during my stay there. I wish to thank Tiago Barreiro, Roger Clay, Ed Copeland, Elisa Di Pietro, Chris Kochanek, David Mota, Nelson Nunes, Ray Protheroe, Francesca Rosati, Luis Ureña-López, and Ioav Waga, for useful cliscussions' Finally, I wish to thank my family and friends, especially Jian Haur, Kwong, and Ling, for their love and encouragement. Chapter 1 Introduction 1.1 Quintessence as Missing Energy The standard cosmological model is the h,ot big bang model which is a homogeneous and isotropic universe with evolution governed by the Friedmann equations obtainecl from General Relativity. Its main constituents can be described by matter and radiation fluids, and its kinematic properties match those we observe in the real universe. It has achieved the foliowing successes: the predictions of light element abundances produced during cosmological nucleosynthesis agree with observations; the cosmic microwave background is naturally explained as a relic of the initial hot thermal phase; it accounts naturally for the expansion of the universe; it provides a framework within which one can understand the formation of galaxies and other cosmic structures. The inflationary uniuerse scenario is designed to address several questions left unanswered by the standard hot big bang scenario. It is an early phase of exponen- tial expansion of the universe that drastically changes the past light cone, removing the horizon problem, and diluting unwanted relics to such very low densities that they are close to unobservable. There exist many different versions of the infla- tionary universe. The first inflationary model was formulated by Guth [1] in 1981, although many of his ideas had been presented previously by Starobinsky [2] in 1979. Although Guth's model proved problematic, improved models with alternative in- flationary mechanisms were subsequently proposed by Albrecht and Steinhardt [3] and by Linde [4] and there are no\n/ very many variants of inflationary cosmologies [5]. The inflationary universe scenario predicts that the universe is spatially flat and 1 that the total energy density of the universe, ptot, is very nearly equal to the critical clensity, p"=3Hll8nG - L7 xl0-2n g "--t, where Ils is the current value of the Hubble parameter and G is Newton's gravitational constant. The cosmic microwave background (CMB), first discovered by Penzias and Wil- son in 1965, is a background radiation whose spectrum is very nearly that of the blackbody radiation of temperature - 2.7 K. The angular power spectrum of tem- perature anisotropy in the CMB is a powerful probe of the content and na,ture of the universe. Current measurements of the CMB anisotropy include the balloon- borne experiments BOOMERanG [6] and MAXIMA [7], and the Degree Angular Scale Interferometer (DASI) [8]. These experiments have measured the CMB power spectrum ovel a wide range of spherical harmonic multipoles. New constraints have been set on a seven-climensional space of cosmological parameters within the class of inflationary adiabatic models using only weakly restrictive prior probabilities. Tlre total energy content of the universe is detelmined to be f)¿o¿ = puf P" - 1.0, consistent with the prediction of the inflationary universe scenario. At the same time, there is growing observational evidence that the total matter density p^ of the universe is significantly less than the critical density (see, e.g., refs. [9]-[14] and references therein). Constraints on the matter energy density of the universe can be derived, €.8., from galaxy cluster abundances [12, 13], and large- scale structure [14]. These tests are consistent with f]- = P,nlP. - 0.3. Primordial nucleosynthesis constraints require a significant ploportion of this matter density to be non-baryonic, which would indicate that the univelse contains significant "dark matter" of a form which is not readily found in the laboratory. If these results on the density of clumped matter hold and measurements of the CMB anisotropy establish that the universe is spatially flat, then there must be another contribution to the energy density of the universe, which cloes not clump gravitationally in the way that ordinary matter does. It is a "smooth" component, which is now often called a "dark energy" to contrast it from gravitationally clumped dark matter. Furthermore, the unexpected faintness of type Ia supernovae (SNe Ia) at recl- shift z - 0.5 provides the most direct evidence that the expansion of the Universe is accelerating [15, 16]. This conclusion is supported by measurements of the charac- teristic angular scale of fluctuations in the CMB which reveal a total energy densìty well in excess of the fraction attributed to gravitating mass 17,77,18]. Such an ex- pansion is quite unexpected, violating Newtonian intuition and requiring some form 2 of universal repulsion to counteract the decelerating effects of normal matter and radiation. This suggests that the missing energy should possess negative pressure, : 7s 10,, and equation-of-state, u plp < 0. To date the evidence for a universe which is accelerating at the present epoch is based on renormalizing the peak luminosities of SNe Ia according to empirical relations, so-called "Phillips relations" [19] [25], which have been observed between the peak iuminosities and decay times of SNe Ia at low redshifts. Thus the evidence for acceleration is not fully conclusive, although the bulk of observational checks do seem to support the acceleration hypothesis [26]. It is one of the aims of this thesis to explore the extent to which current data and possible future satellite data can dis- tinguish the signature of cosmological acceleration from other possible explanations, such as an evolution of the SNe Ia sources. Whatever doubts one might have about the evidence for cosmological accelera- tion, among theoretical cosmologists the possibility has seen a great resurgence of interest in models involving the cosmological constant,,A., a vacuum energy density wlrich is spatially uniform and time-independent, with wL : -r' The cosmolog- ical constant was first introduced by Einstein fol the purpose of allowing a static universe with the repulsive cosmological constant delicately balancing the gravita- tional attraction of rnatter [27].