On the limit cycle of an inflationary universe (*)
|
|
- Myron Simpson
- 5 years ago
- Views:
Transcription
1 IL NUOVO CIMENTO VOL. 112 B, N. 6 Giugno 1997 On the limit cycle of an inflationary universe (*) L. SALASNICH (**) Dipartimento di Matematica Pura ed Applicata, Università di Padova Via Marzolo 8, I Padova, Italy INFN, Sezione di Padova - Via Marzolo 8, I Padova, Italy (ricevuto il 25 Settembre 1996; approvato il 26 Novembre 1996) Summary. We study the dynamics of a scalar inflaton field with a symmetric double-well potential and prove rigorously the existence of a limit cycle in its phase space. By using analytical and numerical arguments we show that the limit cycle is stable and give an analytical formula for its period. PACS Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.). PACS Lm Nonlinear or nonlocal theories and models. The nonlinear and chaotic behaviour of classical field theories is currently subject of intensive research [1-3] and, in this respect, it is of great interest to investigate the existence and properties of limit cycles, which are inherently nonlinear phenomena [4, 5]. In a previous paper [6] we studied the stability of a scalar inflaton field with a symmetric double-well self-energy. We showed that the value of the inflaton field in the vacuum is a bifurcation parameter which changes the phase space structure and that for some functional solutions of the Hubble constant the system goes to a limit cycle, i.e. to a periodic orbit. In this paper we analyze the properties of this limit cycle by using analytical and numerical arguments. We show that the limit cycle is unique and stable and give an analytical formula for its period. To solve the three major cosmological problems, i.e. the flatness problem, the homogeneity problem, and the formation of structure problem, it is generally postulated that the universe, at a very early stage after the big bang, exhibited a short period of exponential expansion, the so-called inflationary phase [7-10]. All the inflationary models assume the existence of a scalar field f, the so-called inflation field, (*) The author of this paper has agreed to not receive the proofs for correction. (**) salasnichhmath.unipd.it 873
2 874 L. SALASNICH with Lagrangian [9, 11] (1) L4 1 2 m f m f2v(f), where the potential V(f) depends on the type of inflation model considered. Here we choose a real field but also complex scalar can be used [9]. The scalar field, if minimally coupled to gravity, satisfies the equation (2) pf4f O 13u ȧ a v ḟ2 1 a 2 2 f42 V f, where p is the covariant d Alembertian operator and a is the cosmological scale factor. The parameter G is the gravitational constant (G4M p 22 with ˇ4c41 and M p 41.2Q GeV the Plank mass) and H4 ȧoa is the Hubble constant, which in general is a function of time (Hubble function). We suppose that in the universe there is only the inflaton field, so the Hubble function H is related to the energy density of the field by (3) H 2 1 u k a 4 ȧ 2 v 1 y k 2 a a 4 8pG ḟ ( f) 2 2 1V(f)z. Immediately after the onset of inflation, the cosmological scale factor a grows exponentially [9]. Thus the term 2 foa 2 is generally believed to be negligible and, if the inflaton field is sufficiently uniform (i.e. ḟ 2, ( f) 2 bv(f)), we end up with a classical nonlinear scalar field theory in one dimension: (4) f O 13H(f) ḟ1 V f 40, where the Hubble function H satisfies the equation (5) H 2 4 8pG V(f). 3 The potential V(f) depends on the type of inflation model considered and we choose a symmetric double-well potential (6) V(f) 4 l 4 (f 2 2v 2 ) 2, where 6v are the values of the inflaton field in the vacuum, i.e. the points of minimal energy of the system. One of the main difficulties in constructing models with potentials suitable for inflation is that these potentials must be flat enough to allow a sufficiently long period of inflation [8, 9]. In this respect our model is very schematic but it can be seen as a toy model for classical nonlinear dynamics with the attractive feature that it emerges from inflationary cosmology. Obviously, the complete study of the dynamics of the inflaton may be addressed only by a complete quantum field theory approach able to predict not only the behaviour of the classical value of the inflaton field but also the associated
3 ON THE LIMIT CYCLE OF AN INFLATIONARY UNIVERSE 875 quantum fluctuations. However, the quantization of the inflation scenario is still an open problem [10] (an interesting stochastic approach can be found in [12]). In paper [6] we showed that the inflaton field value in the vacuum v is a bifurcation parameter. If v40 in the phase space there is only one stable fixed point (f40, ḟ40), which is an attractor. Instead for vc0 there are three fixed points: (f40, ḟ40), which is unstable, and (f46v, ḟ40), which are stable. The Hubble function is determined by solving eq. (5). There are four possible continuous solutions: (7) H(f) 46gNf 2 2v 2 N, Fig. 1. The Hubble function vs. time (top) and the phase space trajectory of the inflaton field (bottom); for H(f) 4gNf 2 2v 2 N with g41o2, l43 and v41. Initial conditions: f40 and ḟ 41O2.
4 876 L. SALASNICH but also (8) H(f) 46g(f 2 2v 2 ), where g4k2pglo3 is the friction parameter. The choice of the solution is crucial for the dynamical evolution of the system. By using the Bendixon criterion [13] (discussed in detail in [6]) we obtain that if H(f) 4gNf 2 2v 2 N then the Hubble function does not change sign and we do not find periodic orbits. The 4th-order Runge-Kutta numerical integration [14] of the equations of motion shows that for v c 0 the inflation field approaches one of its two stable fixed-point attractors, and that the Hubble function goes to zero with an oscillatory behaviour (see fig. 1). Instead, if we choose H(f) 4g(f 2 2v 2 ), then the Hubble Fig. 2. The Hubble function vs. time (top) and the phase space trajectory of the inflaton field (bottom); for H(f) 4g(f 2 2v 2 ) with g41o2, l43 and v41. Initial conditions: f40 and ḟ 44.
5 ON THE LIMIT CYCLE OF AN INFLATIONARY UNIVERSE 877 function can change sign and the Bendixon criterion admits for vc0 the existence of a limit cycle. In fact, the numerical calculations plotted in fig. 2 show that a limit cycle exists and that the Hubble function oscillates forever. Now we want analyze in detail the properties of this limit cycle. The equation of motion of the inflaton field with H(f) 4g(f 2 2v 2 ) reads (9) f O 13g(f 2 2v 2 ) ḟ1lf(f 2 2v 2 ) 40. This equation can be written as d yḟ13g f (10) (u 2 2v 2 )duz1lf(f 2 2v 2 ) 40, dt 0 and if we put f (11) F(f) 43 (u 2 2v 2 )du4f(f 2 23v 2 ), G(f)4f(f 2 2v 2 ), 0 and also v4 ḟ1gf(f), we obtain the system (12). / ḟ 4v2gF(f), v. 42lG(f). For systems of this kind the Lienard theorem [15, 16] states that there is a unique and stable limit cycle if the following conditions are satisfied: F(f) is an odd function and F(f) 40 only at f40 and f46a; F(f)E0 for 0 EfEa, F(f) D0 and is increasing for fda; G(f) is an odd function and fg(f) D0 for all fda. It is easy to check that the functions F(f) and G(f) defined by (11) satisfy all the conditions of the Lienard theorem with a4v. The cubic force G(f) tends to reduce any displacement for large NfN, whereas the damping F(f) is negative at small NfN and positive at large NfN. Since small oscillations are pumped up and large oscillations are damped down, it is not surprising that the system tends to settle into a self-sustained oscillation of some intermediate amplitude. Figures 2 and 3 show that both internal and external initial conditions generate trajectories which approach the limit cycle, so we have also a numerical evidence of the stability of the limit cycle. Let us consider a typical trajectory of the Lienard system (12). After the scaling c4lv we obtain (13). F(f)l ḟ 4lkc2 g l /, ċ 42G(f). The cubic nullcline c4 (gol)f(f) is the key to understand the motion [5]. Suppose that lc1 and the initial condition is far from the cubic nullcline, then (13) implies NḟNAO(l)c1; hence the velocity is enormous in the horizontal direction and tiny in the vertical direction, so trajectories move practically horizontally. If the initial condition is above the nullcline then ḟ D 0, thus the trajectory moves sideways toward
6 878 L. SALASNICH Fig. 3. The Hubble function vs. time (top) and the phase space trajectory of the inflaton field (bottom); for H(f) 4g(f 2 2v 2 ) with g41o2, l43 and v41. Initial conditions: f421o2 and ḟ 40. the nullcline. However, once the trajectory gets so close that cc (log) F(f), then the trajectory crosses the nullcline vertically and moves slowly along the backside of the branch until it reaches the knee and can jump sideways again. The period T of the limit cycle is essentially the time required to travel along the two slow branches, since the time spent in the jumps is negligible for large l. By symmetry, the time spent on each branch is the same so we have (14) t B TC2 dt, t A
7 ON THE LIMIT CYCLE OF AN INFLATIONARY UNIVERSE 879 where A and B are the initial and final points on the positive slow branch. To derive an expression for dt we note that on the slow branches with a good approximation cc (gol) F(f) and thus (15) dc dt C g l F 8(f) df dt 43 g l (f 2 2v 2 ) df dt. Since from (13) dcodt42f(f 2 2v 2 ), we obtain (16) dtc23 g l df f, on the slow branches. The slow positive branch begins at f A 42gvOl and ends at f B 4 gvol, hence (17) Because g4k2pglo3 we have t B TC2dtC26 g f Bdf l f C6 g l ln 2. f A t A (18) TC2 ln2o 6pG l. Note that the period is v-independent. In summary, we have proved the existence and stability of a limit cycle in the phase space of a scalar inflaton field f with a symmetric double-well potential V(f) and a friction term in the equation of motion proportional to V(f). Then we have obtained an analytical estimation of the period of the limit cycle. *** The author is greatly indebted to V. R. MANFREDI and M. ROBNIK for many enlightening discussions. R E F E R E N C E S [1] KAWABE T. and OHTA S., Phys. Lett. B, 334 (1994) 127; KAWABE T., Phys. Lett. B, 343 (1995) 225. [2] SALASNICH L., Phys. Rev. D., 52 (1995) 6189; GRAFFI S., MANFREDI V. R. and SALASNICH L., Mod. Phys. Lett. B, 7 (1995) 747. [3] SEGAR J. and SRIRAM M. S., Phys. Rev. D, 53 (1996) [4] NAYFEH A. H. and BALACHANDRAN B., Applied Nonlinear Dynamics (J. Wiley, New York) [5] FARKAS M., Periodic Motion (Springer, Berlin) [6] SALASNICH L., Mod. Phys. Lett. A, 10 (1995) [7] GUTH A. H., Phys. Rev. D, 23 (1981) 347. [8] LINDE A. D., Phys. Lett. B, 108 (1982) 389; 129 (1983) 177. [9] LINDE A. D., Particle Physics and Inflationary Cosmology (Harwood Academic Publishers, London) 1988.
8 880 L. SALASNICH [10] BRANDENBERGER R. H., in SUSSP Proceedings: Physics of the Early Universe, edited by J. A. PEACOCK, A. F. HEAVENS and A. T. DAVES (Institute of Physics Publishing, Bristol) [11] ITZYKSON C. and ZUBER J. B., Quantum Field Theory (McGraw-Hill, New York) [12] BECK C., Nonlinearity, 8 (1995) 423. [13] BENDIXSON I., Acta Math., 24 (1901) 1. [14] Subroutine D02BAF, The NAG Fortran Library, Mark 14, Oxford: NAG Ltd. and USA: NAG Inc. (1990). [15] LIENARD A., Rev. Gen. Electr., 23 (1928) 901. [16] JORDAN D. W. and SMITH P., Nonlinear Ordinary Differential Equations (Oxford University Press, Oxford) 1987.
8 Example 1: The van der Pol oscillator (Strogatz Chapter 7)
8 Example 1: The van der Pol oscillator (Strogatz Chapter 7) So far we have seen some different possibilities of what can happen in two-dimensional systems (local and global attractors and bifurcations)
More informationStructures in the early Universe. Particle Astrophysics chapter 8 Lecture 4
Structures in the early Universe Particle Astrophysics chapter 8 Lecture 4 overview Part 1: problems in Standard Model of Cosmology: horizon and flatness problems presence of structures Part : Need for
More informationOn the Torus Quantization of Two Anyons with Coulomb Interaction in a Magnetic Field
Preprint DFPD/97/TH/15 On the Torus Quantization of Two Anyons with Coulomb Interaction in a Magnetic Field Luca Salasnich 1 Dipartimento di Matematica Pura ed Applicata Università di Padova, Via Belzoni
More informationINFLATION. - EARLY EXPONENTIAL PHASE OF GROWTH OF SCALE FACTOR (after T ~ TGUT ~ GeV)
INFLATION - EARLY EXPONENTIAL PHASE OF GROWTH OF SCALE FACTOR (after T ~ TGUT ~ 10 15 GeV) -Phenomenologically similar to Universe with a dominant cosmological constant, however inflation needs to end
More informationCOSMIC INFLATION AND THE REHEATING OF THE UNIVERSE
COSMIC INFLATION AND THE REHEATING OF THE UNIVERSE Francisco Torrentí - IFT/UAM Valencia Students Seminars - December 2014 Contents 1. The Friedmann equations 2. Inflation 2.1. The problems of hot Big
More informationA Magnetized Kantowski-Sachs Inflationary Universe in General Relativity
Bulg. J. Phys. 37 (2010) 144 151 A Magnetized Kantowski-Sachs Inflationary Universe in General Relativity S.D. Katore PG Department of Mathematics, SGB Amravati University, Amravati, India Received 10
More informationIntroduction to Inflation
Introduction to Inflation Miguel Campos MPI für Kernphysik & Heidelberg Universität September 23, 2014 Index (Brief) historic background The Cosmological Principle Big-bang puzzles Flatness Horizons Monopoles
More informationarxiv:gr-qc/ v3 17 Jul 2003
REGULAR INFLATIONARY COSMOLOGY AND GAUGE THEORIES OF GRAVITATION A. V. Minkevich 1 Department of Theoretical Physics, Belarussian State University, av. F. Skoriny 4, 0050, Minsk, Belarus, phone: +37517095114,
More informationCosmic Bubble Collisions
Outline Background Expanding Universe: Einstein s Eqn with FRW metric Inflationary Cosmology: model with scalar field QFTà Bubble nucleationà Bubble collisions Bubble Collisions in Single Field Theory
More informationThe Theory of Inflationary Perturbations
The Theory of Inflationary Perturbations Jérôme Martin Institut d Astrophysique de Paris (IAP) Indian Institute of Technology, Chennai 03/02/2012 1 Introduction Outline A brief description of inflation
More informationExact Inflationary Solution. Sergio del Campo
Exact Inflationary Solution Sergio del Campo Instituto de Física Pontificia Universidad Católica de Valparaíso Chile I CosmoSul Rio de Janeiro, 1 al 5 de Agosto, 2011 Inflation as a paradigm. Models Slow-roll
More informationStructures in the early Universe. Particle Astrophysics chapter 8 Lecture 4
Structures in the early Universe Particle Astrophysics chapter 8 Lecture 4 overview problems in Standard Model of Cosmology: horizon and flatness problems presence of structures Need for an exponential
More informationBianchi Type VIII Inflationary Universe with Massless Scalar Field in General Relativity
August 05 Volume 6 Issue 8 pp. 679-68 Bali,. & Swati, Bianchi Type VIII Inflationary Universe with Massless Scalar Field in General elativity Bianchi Type VIII Inflationary Universe with Massless Scalar
More informationInflation. By The amazing sleeping man, Dan the Man and the Alices
Inflation By The amazing sleeping man, Dan the Man and the Alices AIMS Introduction to basic inflationary cosmology. Solving the rate of expansion equation both analytically and numerically using different
More informationThe Concept of Inflation
The Concept of Inflation Introduced by Alan Guth, circa 1980, to provide answers to the following 5 enigmas: 1. horizon problem. How come the cosmic microwave background radiation is so uniform in very
More informationDark inflation. Micha l Artymowski. Jagiellonian University. January 29, Osaka University. arxiv:
Dark inflation Micha l Artymowski Jagiellonian University January 29, 2018 Osaka University arxiv:1711.08473 (with Olga Czerwińska, M. Lewicki and Z. Lalak) Cosmic microwave background Cosmic microwave
More informationBianchi Type-III Inflationary Universe with Constant Deceleration Parameter in General Relativity
Bulg. J. Phys. 38 2011 139 1 Bianchi Type-III Inflationary Universe with Constant Deceleration Parameter in General Relativity S.D. Katore Department of Mathematics, S.G.B. Amravati University, Amravati
More informationCosmic Inflation Tutorial
Cosmic Inflation Tutorial Andreas Albrecht Center for Quantum Mathematics and Physics (QMAP) and Department of Physics UC Davis Simons Workshop on Quantum Information in Cosmology Niels Bohr Institute
More informationarxiv: v1 [gr-qc] 23 Jul 2010
Primordial inflation from gravity s rainbow arxiv:1007.4087v1 [gr-qc] 23 Jul 2010 Christian Corda June 27, 2018 Associazione Scientifica Galileo Galilei, Via Bruno Buozzi 47-59100 PRATO, Italy E-mail address:
More informationCosmology holography the brain and the quantum vacuum. Antonio Alfonso-Faus. Departamento de Aerotécnia. Madrid Technical University (UPM), Spain
Cosmology holography the brain and the quantum vacuum Antonio Alfonso-Faus Departamento de Aerotécnia Madrid Technical University (UPM), Spain February, 2011. E-mail: aalfonsofaus@yahoo.es Abstract: Cosmology,
More informationEternal Inflation Theory, the Multiverse, and Cosmology. Inflation and inflationary cosmology have become a big part of the popular
Scientific Paper I wrote this paper about inflation theory for Astronomy 317: Our Universe, the Final Frontier (Galaxies and Cosmology). I researched a cosmological model and presented the evidence that
More informationGuido D Amico Center for Cosmology and Particle Physics New York University. Unwinding Inflation
Guido D Amico Center for Cosmology and Particle Physics New York University Unwinding Inflation New Lights in Cosmology from the CMB ICTP Trieste, Summer 2013 with Roberto Gobbetti, Matthew Kleban, Marjorie
More informationCould the Higgs Boson be the Inflaton?
Could the Higgs Boson be the Inflaton? Michael Atkins Phys.Lett. B697 (2011) 37-40 (arxiv:1011.4179) NExT Meeting March 2012, Sussex Outline Why inflation? The Higgs as the inflaton Unitarity and Higgs
More informationarxiv: v2 [hep-ph] 5 Feb 2015
Spiral Inflation with Coleman-Weinberg Potential FTU-14-12-31 IFIC-14-86 Gabriela Barenboim and Wan-Il Park Departament de Física Teòrica and IFIC, Universitat de alència-csic, E-46100, Burjassot, Spain
More informationEffects of the field-space metric on Spiral Inflation
Effects of the field-space metric on Spiral Inflation Josh Erlich College of William & Mary digitaldante.columbia.edu Miami 2015 December 20, 2015 The Cosmic Microwave Background Planck collaboration Composition
More informationInflation Scheme Derived from Universal Wave Function Interpretation of String Theory
Journal of Physical Science and Application 7 (4) (2017) 33-37 doi: 10.17265/2159-5348/2017.04.004 D DAVID PUBLISHING Inflation Scheme Derived from Universal Wave Function Interpretation of String Theory
More informationarxiv:hep-th/ v2 12 Oct 1994
UH-IfA-94/35 SU-ITP-94-13 YITP/U-94-15 hep-th/9405187 REHEATING AFTER INFLATION arxiv:hep-th/9405187v2 12 Oct 1994 Lev Kofman Institute for Astronomy, University of Hawaii, 2680 Woodlawn Dr., Honolulu,
More informationXIII. The Very Early Universe and Inflation. ASTR378 Cosmology : XIII. The Very Early Universe and Inflation 171
XIII. The Very Early Universe and Inflation ASTR378 Cosmology : XIII. The Very Early Universe and Inflation 171 Problems with the Big Bang The Flatness Problem The Horizon Problem The Monopole (Relic Particle)
More informationNumber of limit cycles of the Liénard equation
PHYSICAL REVIEW E VOLUME 56, NUMBER 4 OCTOBER 1997 Number of limit cycles of the Liénard equation Hector Giacomini and Sébastien Neukirch Laboratoire de Mathématiques et Physique Théorique, CNRS UPRES
More informationNon-singular quantum cosmology and scale invariant perturbations
th AMT Toulouse November 6, 2007 Patrick Peter Non-singular quantum cosmology and scale invariant perturbations Institut d Astrophysique de Paris GRεCO AMT - Toulouse - 6th November 2007 based upon Tensor
More informationStrong-coupling scale and frame-dependence of the initial conditions for chaotic inflation in models with modified (coupling to) gravity
arxiv:1607.05268v1 [gr-qc] 17 Jul 2016 Strong-coupling scale and frame-dependence of the initial conditions for chaotic inflation in models with modified (coupling to) gravity Dmitry Gorbunov, Alexander
More informationTheoretical physics. Deterministic chaos in classical physics. Martin Scholtz
Theoretical physics Deterministic chaos in classical physics Martin Scholtz scholtzzz@gmail.com Fundamental physical theories and role of classical mechanics. Intuitive characteristics of chaos. Newton
More informationHolographic Model of Cosmic (P)reheating
Holographic Model of Cosmic (P)reheating Yi-Fu Cai 蔡一夫 University of Science & Technology of China New perspectives on Cosmology, APCTP, Feb 13 th 2017 In collaboration with S. Lin, J. Liu & J. Sun, Based
More informationScale symmetry a link from quantum gravity to cosmology
Scale symmetry a link from quantum gravity to cosmology scale symmetry fluctuations induce running couplings violation of scale symmetry well known in QCD or standard model Fixed Points Quantum scale symmetry
More informationThe Standard Big Bang What it is: Theory that the universe as we know it began billion years ago. (Latest estimate: 13:7 ± 0:2 billion years!) I
The Standard Big Bang What it is: Theory that the universe as we know it began 13-15 billion years ago. (Latest estimate: 13:7 ± 0:2 billion years!) Initial state was a hot, dense, uniform soup of particles
More information4.3 The accelerating universe and the distant future
Discovering Astronomy : Galaxies and Cosmology 46 Figure 55: Alternate histories of the universe, depending on the mean density compared to the critical value. The left hand panel shows the idea graphically.
More informationPatrick Peter. Institut d Astrophysique de Paris Institut Lagrange de Paris. Evidences for inflation constraints on alternatives
Patrick Peter Institut d Astrophysique de Paris Institut Lagrange de Paris Evidences for inflation constraints on alternatives Thanks to Jérôme Martin For his help Planck 2015 almost scale invariant quantum
More informationPREHEATING THE UNIVERSE IN HYBRID INFLATION
PREHEATING THE UNIVERSE IN HYBRID INFLATION JUAN GARCÍA-BELLIDO Theory Division, C.E.R.N., CH-1211 Genève 23, Switzerland One of the fundamental problems of modern cosmology is to explain the origin of
More informationCHAPTER 4 INFLATIONARY MODEL BUILDING. 4.1 Canonical scalar field dynamics. Non-minimal coupling and f(r) theories
CHAPTER 4 INFLATIONARY MODEL BUILDING Essentially, all models are wrong, but some are useful. George E. P. Box, 1987 As we learnt in the previous chapter, inflation is not a model, but rather a paradigm
More informationGalaxies 626. Lecture 3: From the CMBR to the first star
Galaxies 626 Lecture 3: From the CMBR to the first star Galaxies 626 Firstly, some very brief cosmology for background and notation: Summary: Foundations of Cosmology 1. Universe is homogenous and isotropic
More informationWill Planck Observe Gravity Waves?
Will Planck Observe Gravity Waves? Qaisar Shafi Bartol Research Institute Department of Physics and Astronomy University of Delaware in collaboration with G. Dvali, R. K. Schaefer, G. Lazarides, N. Okada,
More informationInflationary Massive Gravity
New perspectives on cosmology APCTP, 15 Feb., 017 Inflationary Massive Gravity Misao Sasaki Yukawa Institute for Theoretical Physics, Kyoto University C. Lin & MS, PLB 75, 84 (016) [arxiv:1504.01373 ]
More informationGravitation et Cosmologie: le Modèle Standard Cours 8: 6 fevrier 2009
Particules Élémentaires, Gravitation et Cosmologie Année 2008-09 Gravitation et Cosmologie: le Modèle Standard Cours 8: 6 fevrier 2009 Le paradigme inflationnaire Homogeneity and flatness problems in HBB
More informationarxiv:gr-qc/ v1 4 Dec 1997
Proof of the cosmic no-hair conjecture for quadratic homogeneous cosmologies arxiv:gr-qc/97106v1 4 Dec 1997 S Cotsakis, J Miritzis Department of Mathematics, University of the Aegean, Karlovassi 8300,
More informationArchaeology of Our Universe YIFU CAI ( 蔡一夫 )
Archaeology of Our Universe YIFU CAI ( 蔡一夫 ) 2013-11-05 Thermal History Primordial era 13.8 billion years by WMAP/NASA Large Scale Structure (LSS) by 2MASS Cosmic Microwave Background (CMB) by ESA/Planck
More informationKey: cosmological perturbations. With the LHC, we hope to be able to go up to temperatures T 100 GeV, age t second
Lecture 3 With Big Bang nucleosynthesis theory and observations we are confident of the theory of the early Universe at temperatures up to T 1 MeV, age t 1 second With the LHC, we hope to be able to go
More informationBianchi Type-VI Inflationary Universe in General Relativity
March 01 Vol. 3 Issue 5 pp. 7-79 Katore S. D. & Chopade B. B. Bianchi Type-VI Inflationary Universe in General Relativity Bianchi Type-VI Inflationary Universe in General Relativity 7 Article Shivdas.
More informationSchool Observational Cosmology Angra Terceira Açores 3 rd June Juan García-Bellido Física Teórica UAM Madrid, Spain
School Observational Cosmology Angra Terceira Açores 3 rd June 2014 Juan García-Bellido Física Teórica UAM Madrid, Spain Outline Lecture 1 Shortcomings of the Hot Big Bang The Inflationary Paradigm Homogeneous
More informationarxiv:hep-th/ v1 12 Mar 2002
UPR-978-T On the Signature of Short Distance Scale in the Cosmic Microwave Background ariv:hep-th/0203113v1 12 Mar 2002 Gary Shiu 1 and Ira Wasserman 2 1 Department of Physics and Astronomy, University
More informationGraceful exit from inflation for minimally coupled Bianchi A scalar field models
Graceful exit from inflation for minimally coupled Bianchi A scalar field models Florian Beyer Reference: F.B. and Leon Escobar (2013), CQG, 30(19), p.195020. University of Otago, Dunedin, New Zealand
More informationSteady-State Cosmology in the Yilmaz Theory of Gravitation
Steady-State Cosmology in the Yilmaz Theory of ravitation Abstract H. E. Puthoff Institute for Advanced Studies at Austin 43 W. Braker Ln., Suite 3 Austin, Texas 78759 Yilmaz has proposed a modification
More informationMASAHIDE YAMAGUCHI. Quantum generation of density perturbations in the early Universe. (Tokyo Institute of Technology)
Quantum generation of density perturbations in the early Universe MASAHIDE YAMAGUCHI (Tokyo Institute of Technology) 03/07/16@Symposium: New Generation Quantum Theory -Particle Physics, Cosmology, and
More informationContents. Part I The Big Bang and the Observable Universe
Contents Part I The Big Bang and the Observable Universe 1 A Historical Overview 3 1.1 The Big Cosmic Questions 3 1.2 Origins of Scientific Cosmology 4 1.3 Cosmology Today 7 2 Newton s Universe 13 2.1
More informationNonminimal coupling and inflationary attractors. Abstract
608.059 Nonminimal coupling and inflationary attractors Zhu Yi, and Yungui Gong, School of Physics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China Abstract We show explicitly
More informationPrimordial Gravity s Breath
EJTP 9, No. 26 (2012) 1 10 Electronic Journal of Theoretical Physics Primordial Gravity s Breath Christian Corda 1 International Institute for Theoretical Physics and Advanced Mathematics Einstein-Galilei,
More informationInflation and the cosmological constant problem
Inflation and the cosmological constant problem Larissa Lorenz Sebastian Sapeta Krzyzowa 18. 8. September 00 Contents Standard model of cosmology and its problems The inflationary paradigm Review of the
More informationCRITICAL SLOWING DOWN AND DEFECT FORMATION M. PIETRONI. INFN - Sezione di Padova, via F. Marzolo 8, Padova, I-35131, ITALY
CRITICAL SLOWING DOWN AND DEFECT FORMATION M. PIETRONI INFN - Sezione di Padova, via F. Marzolo 8, Padova, I-35131, ITALY E-mail: pietroni@pd.infn.it The formation of topological defects in a second order
More informationTriple unification of inflation, dark matter and dark energy
Triple unification of inflation, dark matter and dark energy May 9, 2008 Leonard Susskind, The Anthropic Landscape of String Theory (2003) A. Liddle, A. Ureña-López, Inflation, dark matter and dark energy
More informationLuca Salasnich 2. Dipartimento di Fisica "G. Galilei" dell'universita di Padova, Via Marzolo 8, I Padova, Italy. and.
Chaos and Quantum Chaos in Nuclear Systems 1 Luca Salasnich Dipartimento di Fisica "G. Galilei" dell'universita di Padova, Via Marzolo 8, I 35131 Padova, Italy and Departamento de Fisica Atomica, Molecular
More informationInflation and the Primordial Perturbation Spectrum
PORTILLO 1 Inflation and the Primordial Perturbation Spectrum Stephen K N PORTILLO Introduction The theory of cosmic inflation is the leading hypothesis for the origin of structure in the universe. It
More informationPREHEATING, PARAMETRIC RESONANCE AND THE EINSTEIN FIELD EQUATIONS
PREHEATING, PARAMETRIC RESONANCE AND THE EINSTEIN FIELD EQUATIONS Matthew PARRY and Richard EASTHER Department of Physics, Brown University Box 1843, Providence RI 2912, USA Email: parry@het.brown.edu,
More informationarxiv:astro-ph/ v4 17 Sep 2004
A TIME VARYING STRONG COUPLING CONSTANT AS A MODEL OF INFLATIONARY UNIVERSE arxiv:astro-ph/000904v4 17 Sep 004 N. Chamoun 1,, S. J. Landau 3,4, H. Vucetich 1,3, 1 Departamento de Física, Universidad Nacional
More informationDIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS
DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS Morris W. Hirsch University of California, Berkeley Stephen Smale University of California, Berkeley Robert L. Devaney Boston University
More informationInflation Andrei Linde. Encyclopedia of Astronomy & Astrophysics P. Murdin
eaa.iop.org DOI: 10.1888/0333750888/2135 Inflation Andrei Linde From Encyclopedia of Astronomy & Astrophysics P. Murdin IOP Publishing Ltd 2006 ISBN: 0333750888 Institute of Physics Publishing Bristol
More informationThe Endless Universe: A Brief Introduction 1
The Endless Universe: A Brief Introduction 1 PAUL J. STEINHARDT Professor of Physics, Princeton University The cyclic model of the universe is a radical alternative to standard big bang/inflationary theory
More informationExcluding Black Hole Firewalls with Extreme Cosmic Censorship
Excluding Black Hole Firewalls with Extreme Cosmic Censorship arxiv:1306.0562 Don N. Page University of Alberta February 14, 2014 Introduction A goal of theoretical cosmology is to find a quantum state
More informationResearch Center for the Early Universe (RESCEU) Department of Physics. Jun ichi Yokoyama
Research Center for the Early Universe (RESCEU) Department of Physics Jun ichi Yokoyama time size Today 13.8Gyr Why is Our Universe Big, dark energy Old, and full of structures? galaxy formation All of
More informationPOST-INFLATIONARY HIGGS RELAXATION AND THE ORIGIN OF MATTER- ANTIMATTER ASYMMETRY
POST-INFLATIONARY HIGGS RELAXATION AND THE ORIGIN OF MATTER- ANTIMATTER ASYMMETRY LOUIS YANG ( 楊智軒 ) UNIVERSITY OF CALIFORNIA, LOS ANGELES (UCLA) DEC 27, 2016 NATIONAL TSING HUA UNIVERSITY OUTLINE Big
More informationShortcomings of the inflationary paradigm
Shortcomings of the inflationary paradigm Looking at the Planck results, Steinhardt et all say:! 1) chaotic inflation with V = O(1) does not work! 2) the only remaining models are the ones with V
More informationInflation and the SLAC Theory Group I was a one-year visitor from a postdoc position at Cornell. My research problem (working with Henry Tye
Inflation and the SLAC Theory Group 1979 1980 I was a one-year visitor from a postdoc position at Cornell. My research problem (working with Henry Tye back at Cornell): Why were so few magnetic monopoles
More informationHIGGS INFLATION & VACUUM STABILITY
HIGGS INFLATION & VACUUM STABILITY Javier Rubio based on Phys. Rev. D 92, 083512 F. Bezrukov, J.R., M.Shaposhnikov Outline Could the Higgs field itself be responsible for inflation? 1. Reminder of inflation/
More informationarxiv:hep-th/ v2 29 Nov 2002
Preferred Frame in Brane World Merab GOGBERASHVILI Andronikashvili Institute of Physics 6 Tamarashvili Str., Tbilisi 380077, Georgia (E-mail: gogber@hotmail.com) arxiv:hep-th/0207042v2 29 Nov 2002 Abstract
More informationBianchi Type VI0 Inflationary Universe with Constant Deceleration Parameter and Flat Potential in General Relativity
Advances in Astrophysics, Vol., No., May 7 https://dx.doi.org/.66/adap.7. 67 Bianchi ype VI Inflationary Universe with Constant Deceleration Parameter and Flat Potential in General Relativity Raj Bali
More informationInflationary cosmology from higher-derivative gravity
Inflationary cosmology from higher-derivative gravity Sergey D. Odintsov ICREA and IEEC/ICE, Barcelona April 2015 REFERENCES R. Myrzakulov, S. Odintsov and L. Sebastiani, Inflationary universe from higher-derivative
More informationLECTURE 8: DYNAMICAL SYSTEMS 7
15-382 COLLECTIVE INTELLIGENCE S18 LECTURE 8: DYNAMICAL SYSTEMS 7 INSTRUCTOR: GIANNI A. DI CARO GEOMETRIES IN THE PHASE SPACE Damped pendulum One cp in the region between two separatrix Separatrix Basin
More informationEternal Inflation in Stringy Landscape. and A-word. Andrei Linde
Eternal Inflation in Stringy Landscape and A-word A Andrei Linde Inflationary Multiverse For a long time, people believed in the cosmological principle, which asserted that the universe is everywhere the
More informationDIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS
DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS Morris W. Hirsch University of California, Berkeley Stephen Smale University of California, Berkeley Robert L. Devaney Boston University
More informationarxiv:gr-qc/ v1 19 May 2006
1 A late time acceleration of the universe with two scalar fields : many possibilities arxiv:gr-qc/0605110v1 19 May 2006 Narayan Banerjee and Sudipta Das Relativity and Cosmology Research Centre, Department
More informationInflation and the origin of structure David Wands Institute of Cosmology and Gravitation University of Portsmouth
Cody Astronomical Society 7 th December 2011 Inflation and the origin of structure David Wands Institute of Cosmology and Gravitation University of Portsmouth outline of my talk: large-structure in the
More informationCosmic Inflation Lecture 16 - Monday Mar 10
Physics 224 Spring 2008 Origin and Evolution of the Universe Cosmic Inflation Lecture 16 - Monday Mar 10 Joel Primack University of California, Santa Cruz Outline L15 L16 WMAP 5-year Data and Papers Released
More informationConnecting Quarks to the Cosmos
Connecting Quarks to the Cosmos Institute for Nuclear Theory 29 June to 10 July 2009 Inflationary Cosmology II Michael S. Turner Kavli Institute for Cosmological Physics The University of Chicago Michael
More informationVanishing Dimensions in Four Dimensional Cosmology with Nonminimal Derivative Coupling of Scalar Field
Advanced Studies in Theoretical Physics Vol. 9, 2015, no. 9, 423-431 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2015.5234 Vanishing Dimensions in Four Dimensional Cosmology with Nonminimal
More informationRolling down solution in a simple mechanical model
P R A Y A S Students Journal of Physics c Indian Association of Physics Teachers Rolling down solution in a simple mechanical model Second year, School Of Physical Sciences, NISER, Bhubaneswar 75 005 Abstract.
More informationGRAVITATIONAL WAVES AND THE END OF INFLATION. Richard Easther (Yale)
GRAVITATIONAL WAVES AND THE END OF INFLATION Richard Easther (Yale) OUTLINE Inflation: a reminder Ending inflation: Parametric resonance / preheating [SKIP: technical calculation] Gravitational wave generation
More informationGravitational waves from the early Universe
Gravitational waves from the early Universe Part 2 Sachiko Kuroyanagi (Nagoya University) 26 Aug 2017 Summer Institute 2017 GWs from inflation Inflation Accelerated expansion in the early Universe Solves
More informationDark inflation. Micha l Artymowski. Jagiellonian University. December 12, 2017 COSPA arxiv:
Dark inflation Micha l Artymowski Jagiellonian University December 12, 2017 COSPA 2017 arxiv:1711.08473 (with Olga Czerwińska, M. Lewicki and Z. Lalak) Cosmic microwave background Cosmic microwave background
More informationCurrent status of inflationary cosmology. Gunma National college of Technology,Japan
Current status of inflationary cosmology Shinji Tsujikawa Gunma National college of Technology,Japan Bright side of the world Recent observations have determined basic cosmological parameters in high precisions.
More informationNon Linear Dynamics in Einstein-Friedman Equations
Non Linear Dynamics in Einstein-Friedman Equations Usman Naseer 2012-10-0054 May 15, 2011 Abstract Einstein-Friedman equations for the dynamics of a spatially homogenous and isotropic universe are rederived
More informationFrom time series to superstatistics
From time series to superstatistics Christian Beck School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E 4NS, United Kingdom Ezechiel G. D. Cohen The Rockefeller University,
More informationA873: Cosmology Course Notes. VII. Inflation
Readings VII. Inflation Alan Guth s Inflationary Universe paper (Phys Rev D, Vol. 23, p. 347, 1981) is a classic, well worth reading. The basics are well covered by Ryden, Chapter 11. For more physics
More informationarxiv:astro-ph/ v1 28 Apr 2005 MIRROR WORLD AND AXION: RELAXING COSMOLOGICAL BOUNDS
International Journal of Modern Physics A c World Scientific Publishing Company arxiv:astro-ph/0504636v1 28 Apr 2005 MIRROR WORLD AND AXION: RELAXING COSMOLOGICAL BOUNDS MAURIZIO GIANNOTTI Dipartimento
More informationNON-STATIONARY RESONANCE DYNAMICS OF THE HARMONICALLY FORCED PENDULUM
CYBERNETICS AND PHYSICS, VOL. 5, NO. 3, 016, 91 95 NON-STATIONARY RESONANCE DYNAMICS OF THE HARMONICALLY FORCED PENDULUM Leonid I. Manevitch Polymer and Composite Materials Department N. N. Semenov Institute
More information1 Introduction QUANTUM BIRTH OF A HOT UNIVERSE
QUANTUM BIRTH OF A HOT UNIVERSE I.G. Dymnikova Institute of Mathematics, Informatics and Physics, University of Olsztyn, Żo lnierska 4, 0-56, Olsztyn, Poland; e-mail: irina@matman.uwn.edu.pl M.L. Fil chenkov
More informationThe Evolving Cosmological Constant (Problem)
The Evolving Cosmological Constant (Problem) N. Itzhaki, (PU) PiPT 2006 Outline The CC problem: then and now (experimental hints). Abbott s model (85). Abbott s model string landscape + anthropic principle.
More informationZhong-Zhi Xianyu (CMSA Harvard) Tsinghua June 30, 2016
Zhong-Zhi Xianyu (CMSA Harvard) Tsinghua June 30, 2016 We are directly observing the history of the universe as we look deeply into the sky. JUN 30, 2016 ZZXianyu (CMSA) 2 At ~10 4 yrs the universe becomes
More informationClosed Universes, de Sitter Space and Inflation
Closed Universes, de Sitter Space and Inflation Chris Doran Cavendish Laboratory Based on astro-ph/0307311 by Lasenby and Doran The Cosmological Constant Dark energy responsible for around 70% of the total
More informationPhysics 133: Extragalactic Astronomy and Cosmology. Week 8
Physics 133: Extragalactic Astronomy and Cosmology Week 8 Outline for Week 8 Primordial Nucleosynthesis Successes of the standard Big Bang model Olbers paradox/age of the Universe Hubble s law CMB Chemical/Physical
More informationCosmological Axion Problem in Chaotic Inflationary Universe
ICRR-Report-374-96-25 UT-758 hep-ph/9608405 Cosmological Axion Problem in Chaotic Inflationary Universe S. Kasuya a, M. Kawasaki a and T. Yanagida b a Institute for Cosmic Ray Research, University of Tokyo,
More informationThe Big Crunch/Big Bang Transition. 1. Measure for inflation 2. Passing through singularities - no beginning proposal
The Big Crunch/Big Bang Transition Neil Turok, Perimeter Institute 1. Measure for inflation 2. Passing through singularities - no beginning proposal 2 inflation * initial conditions * fine-tuned potentials
More informationarxiv: v1 [nlin.cd] 20 Jul 2010
Invariant manifolds of the Bonhoeffer-van der Pol oscillator arxiv:1007.3375v1 [nlin.cd] 20 Jul 2010 R. Benítez 1, V. J. Bolós 2 1 Departamento de Matemáticas, Centro Universitario de Plasencia, Universidad
More information