Different Horizons in Cosmology
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Inflation and the Theory of Cosmological Perturbations
Inflation and the Theory of Cosmological Perturbations A. Riotto INFN, Sezione di Padova, via Marzolo 8, I-35131, Padova, Italy. Abstract These lectures provide a pedagogical introduction to inflation and the theory of cosmological per- turbations generated during inflation which are thought to be the origin of structure in the universe. Lectures given at the: Summer School on arXiv:hep-ph/0210162v2 30 Jan 2017 Astroparticle Physics and Cosmology Trieste, 17 June - 5 July 2002 1 Notation A few words on the metric notation. We will be using the convention (−; +; +; +), even though we might switch time to time to the other option (+; −; −; −). This might happen for our convenience, but also for pedagogical reasons. Students should not be shielded too much against the phenomenon of changes of convention and notation in books and articles. Units We will adopt natural, or high energy physics, units. There is only one fundamental dimension, energy, after setting ~ = c = kb = 1, [Energy] = [Mass] = [Temperature] = [Length]−1 = [Time]−1 : The most common conversion factors and quantities we will make use of are 1 GeV−1 = 1:97 × 10−14 cm=6:59 × 10−25 sec, 1 Mpc= 3.08×1024 cm=1.56×1038 GeV−1, 19 MPl = 1:22 × 10 GeV, −1 −1 −42 H0= 100 h Km sec Mpc =2.1 h × 10 GeV, 2 −29 −3 2 4 −3 2 −47 4 ρc = 1:87h · 10 g cm = 1:05h · 10 eV cm = 8:1h × 10 GeV , −13 T0 = 2:75 K=2.3×10 GeV, 2 Teq = 5:5(Ω0h ) eV, Tls = 0:26 (T0=2:75 K) eV. -
Time-Dependent Cosmological Interpretation of Quantum Mechanics Emmanuel Moulay
Time-dependent cosmological interpretation of quantum mechanics Emmanuel Moulay To cite this version: Emmanuel Moulay. Time-dependent cosmological interpretation of quantum mechanics. 2016. hal- 01169099v2 HAL Id: hal-01169099 https://hal.archives-ouvertes.fr/hal-01169099v2 Preprint submitted on 18 Nov 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Time-dependent cosmological interpretation of quantum mechanics Emmanuel Moulay∗ Abstract The aim of this article is to define a time-dependent cosmological inter- pretation of quantum mechanics in the context of an infinite open FLRW universe. A time-dependent quantum state is defined for observers in sim- ilar observable universes by using the particle horizon. Then, we prove that the wave function collapse of this quantum state is avoided. 1 Introduction A new interpretation of quantum mechanics, called the cosmological interpre- tation of quantum mechanics, has been developed in order to take into account the new paradigm of eternal inflation [1, 2, 3]. Eternal inflation can lead to a collection of infinite open Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) bub- ble universes belonging to a multiverse [4, 5, 6]. This inflationary scenario is called open inflation [7, 8]. -
Primordial Gravitational Waves and Cosmology Arxiv:1004.2504V1
Primordial Gravitational Waves and Cosmology Lawrence Krauss,1∗ Scott Dodelson,2;3;4 Stephan Meyer 3;4;5;6 1School of Earth and Space Exploration and Department of Physics, Arizona State University, PO Box 871404, Tempe, AZ 85287 2Center for Particle Astrophysics, Fermi National Laboratory PO Box 500, Batavia, IL 60510 3Department of Astronomy and Astrophysics 4Kavli Institute of Cosmological Physics 5Department of Physics 6Enrico Fermi Institute University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637 ∗To whom correspondence should be addressed; E-mail: [email protected]. The observation of primordial gravitational waves could provide a new and unique window on the earliest moments in the history of the universe, and on possible new physics at energies many orders of magnitude beyond those acces- sible at particle accelerators. Such waves might be detectable soon in current or planned satellite experiments that will probe for characteristic imprints in arXiv:1004.2504v1 [astro-ph.CO] 14 Apr 2010 the polarization of the cosmic microwave background (CMB), or later with di- rect space-based interferometers. A positive detection could provide definitive evidence for Inflation in the early universe, and would constrain new physics from the Grand Unification scale to the Planck scale. 1 Introduction Observations made over the past decade have led to a standard model of cosmology: a flat universe dominated by unknown forms of dark energy and dark matter. However, while all available cosmological data are consistent with this model, the origin of neither the dark energy nor the dark matter is known. The resolution of these mysteries may require us to probe back to the earliest moments of the big bang expansion, but before a period of about 380,000 years, the universe was both hot and opaque to electromagnetic radiation. -
Evolution of the Cosmological Horizons in a Concordance Universe
Evolution of the Cosmological Horizons in a Concordance Universe Berta Margalef–Bentabol 1 Juan Margalef–Bentabol 2;3 Jordi Cepa 1;4 [email protected] [email protected] [email protected] 1Departamento de Astrofísica, Universidad de la Laguna, E-38205 La Laguna, Tenerife, Spain: 2Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, E-28040 Madrid, Spain. 3Facultad de Ciencias Físicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain. 4Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain. Abstract The particle and event horizons are widely known and studied concepts, but the study of their properties, in particular their evolution, have only been done so far considering a single state equation in a deceler- ating universe. This paper is the first of two where we study this problem from a general point of view. Specifically, this paper is devoted to the study of the evolution of these cosmological horizons in an accel- erated universe with two state equations, cosmological constant and dust. We have obtained closed-form expressions for the horizons, which have allowed us to compute their velocities in terms of their respective recession velocities that generalize the previous results for one state equation only. With the equations of state considered, it is proved that both velocities remain always positive. Keywords: Physics of the early universe – Dark energy theory – Cosmological simulations This is an author-created, un-copyedited version of an article accepted for publication in Journal of Cosmology and Astroparticle Physics. IOP Publishing Ltd/SISSA Medialab srl is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. -
AST4220: Cosmology I
AST4220: Cosmology I Øystein Elgarøy 2 Contents 1 Cosmological models 1 1.1 Special relativity: space and time as a unity . 1 1.2 Curvedspacetime......................... 3 1.3 Curved spaces: the surface of a sphere . 4 1.4 The Robertson-Walker line element . 6 1.5 Redshifts and cosmological distances . 9 1.5.1 Thecosmicredshift . 9 1.5.2 Properdistance. 11 1.5.3 The luminosity distance . 13 1.5.4 The angular diameter distance . 14 1.5.5 The comoving coordinate r ............... 15 1.6 TheFriedmannequations . 15 1.6.1 Timetomemorize! . 20 1.7 Equationsofstate ........................ 21 1.7.1 Dust: non-relativistic matter . 21 1.7.2 Radiation: relativistic matter . 22 1.8 The evolution of the energy density . 22 1.9 The cosmological constant . 24 1.10 Some classic cosmological models . 26 1.10.1 Spatially flat, dust- or radiation-only models . 27 1.10.2 Spatially flat, empty universe with a cosmological con- stant............................ 29 1.10.3 Open and closed dust models with no cosmological constant.......................... 31 1.10.4 Models with more than one component . 34 1.10.5 Models with matter and radiation . 35 1.10.6 TheflatΛCDMmodel. 37 1.10.7 Models with matter, curvature and a cosmological con- stant............................ 40 1.11Horizons.............................. 42 1.11.1 Theeventhorizon . 44 1.11.2 Theparticlehorizon . 45 1.11.3 Examples ......................... 46 I II CONTENTS 1.12 The Steady State model . 48 1.13 Some observable quantities and how to calculate them . 50 1.14 Closingcomments . 52 1.15Exercises ............................. 53 2 The early, hot universe 61 2.1 Radiation temperature in the early universe . -
19. Big-Bang Cosmology 1 19
19. Big-Bang cosmology 1 19. BIG-BANG COSMOLOGY Revised September 2009 by K.A. Olive (University of Minnesota) and J.A. Peacock (University of Edinburgh). 19.1. Introduction to Standard Big-Bang Model The observed expansion of the Universe [1,2,3] is a natural (almost inevitable) result of any homogeneous and isotropic cosmological model based on general relativity. However, by itself, the Hubble expansion does not provide sufficient evidence for what we generally refer to as the Big-Bang model of cosmology. While general relativity is in principle capable of describing the cosmology of any given distribution of matter, it is extremely fortunate that our Universe appears to be homogeneous and isotropic on large scales. Together, homogeneity and isotropy allow us to extend the Copernican Principle to the Cosmological Principle, stating that all spatial positions in the Universe are essentially equivalent. The formulation of the Big-Bang model began in the 1940s with the work of George Gamow and his collaborators, Alpher and Herman. In order to account for the possibility that the abundances of the elements had a cosmological origin, they proposed that the early Universe which was once very hot and dense (enough so as to allow for the nucleosynthetic processing of hydrogen), and has expanded and cooled to its present state [4,5]. In 1948, Alpher and Herman predicted that a direct consequence of this model is the presence of a relic background radiation with a temperature of order a few K [6,7]. Of course this radiation was observed 16 years later as the microwave background radiation [8]. -
Contribution Prospectives 2020 Cosmology with Gravitational Waves
Contribution Prospectives 2020 Cosmology with Gravitational Waves Principal author: D.A. Steer Co-authors: V. Vennin, D. Langlois, K. Noui, F. Nitti, E. Kiritsis, M. Khlopov, E. Huguet (Theory group at APC) Virgo groups at APC, ARTEMIS, ILM, IPHC, IP2I, LAL, LAPP and LKB Abstract Gravitational waves, which were detected directly for the first time by the LIGO-Virgo scientific collaboration, can be used to probe early and late-time cosmology. In this letter we present different aspects of research in the areas of cosmology which we believe will be of crucial importance in the next 10 years: 1) measurements of cosmological parameters with GWs; 2) early universe cosmology with GWs. This letter is supplemented by the companion \Testing General Relativity and modified gravity with Gravitational Waves". Overview The Standard Cosmological model, also referred as ΛCDM, is a highly successful model of the universe, whose predictions have passed many precise observational tests in both the early and late-time universe. Despite that, there remain a number of open questions. One, for instance, pertains to the expansion rate of the universe today | namely the Hubble constant H0 | which is a fundamental parameter of the ΛCDM. Indeed, today there is a 4:2σ discrepancy between −1 −1 the value of H0 = 66:93 ± 0:62 ; kmMpc s inferred by the Planck collaboration from Cosmic −1 −1 Microwave Background (CMB) and the value of H0 = 73:5 ± 1:4 kmMpc s measured from Type Ia Supernovae. A recent improvement of the latter estimation method (using extended HST observations of Cepheids in the LMC) leads to the even larger discrepancy of 4:4σ [1]. -
A New Scale of Particle Mass in a Fractal Universe with A
A new scale of particle mass in a fractal universe with a cosmological constant Scott Funkhouser(1) and Nicola Pugno(2) (1)Dept of Physics, the Citadel, 171 Moultrie St. Charleston, SC, 29409 (2)Department of Structural Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy ABSTRACT Considerations of the total action of a fractal universe with a cosmological constant lead to a new microscopic mass scale that is a function of the fractal dimension and fundamental parameters. For a universe with a fractal dimension D=2 the model predicts a mass that is of order near the nucleon mass. A scaling law between the cosmological constant and the mass of the nucleon follows that is identical to a known scaling law originating with Zel’dovich. This analysis also provides new limitations on the possible nature of dark matter in a fractal universe with a cosmological constant. 1. A new scale of mass Consider a system that is well represented by a fractal structure with dimension D. Let the total mass M of the system be dominated by some number N of a certain species of particle whose mass is m, so that N≈M/m. The total action A of the fractal system must be related to the Planck quantum of action h by [1,2] M ( D +1)/ D A ≈ hN (D +1)/ D ≈ h . (1) m Certain prominent features of the universe exhibit strong signatures of a fractal structure. Insomuch as the universe may be represented by a fractal, (1) should relate the total cosmic action to the parameters of the dominant species of massive particles in the universe [2]. -
Big Bang Nucleosynthesis Finally, Relative Abundances Are Sensitive to Density of Normal (Baryonic Matter)
Big Bang Nucleosynthesis Finally, relative abundances are sensitive to density of normal (baryonic matter) Thus Ωb,0 ~ 4%. So our universe Ωtotal ~1 with 70% in Dark Energy, 30% in matter but only 4% baryonic! Case for the Hot Big Bang • The Cosmic Microwave Background has an isotropic blackbody spectrum – it is extremely difficult to generate a blackbody background in other models • The observed abundances of the light isotopes are reasonably consistent with predictions – again, a hot initial state is the natural way to generate these • Many astrophysical populations (e.g. quasars) show strong evolution with redshift – this certainly argues against any Steady State models The Accelerating Universe Distant SNe appear too faint, must be further away than in a non-accelerating universe. Perlmutter et al. 2003 Riese 2000 Outstanding problems • Why is the CMB so isotropic? – horizon distance at last scattering << horizon distance now – why would causally disconnected regions have the same temperature to 1 part in 105? • Why is universe so flat? – if Ω is not 1, Ω evolves rapidly away from 1 in radiation or matter dominated universe – but CMB analysis shows Ω = 1 to high accuracy – so either Ω=1 (why?) or Ω is fine tuned to very nearly 1 • How do structures form? – if early universe is so very nearly uniform Astronomy 422 Lecture 22: Early Universe Key concepts: Problems with Hot Big Bang Inflation Announcements: April 26: Exam 3 April 28: Presentations begin Astro 422 Presentations: Thursday April 28: 9:30 – 9:50 _Isaiah Santistevan__________ 9:50 – 10:10 _Cameron Trapp____________ 10:10 – 10:30 _Jessica Lopez____________ Tuesday May 3: 9:30 – 9:50 __Chris Quintana____________ 9:50 – 10:10 __Austin Vaitkus___________ 10:10 – 10:30 __Kathryn Jackson__________ Thursday May 5: 9:30 – 9:50 _Montie Avery_______________ 9:50 – 10:10 _Andrea Tallbrother_________ 10:10 – 10:30 _Veronica Dike_____________ 10:30 – 10:50 _Kirtus Leyba________________________ Send me your preference. -
(Ns-Tp430m) by Tomislav Prokopec Part III: Cosmic Inflation
1 Lecture notes on Cosmology (ns-tp430m) by Tomislav Prokopec Part III: Cosmic Inflation In this and in the subsequent chapters we use natural units, that is we set to unity the Planck constant, ~ = 1, the speed of light, c = 1, and the Boltzmann constant, kB = 1. We do however maintain a dimensionful gravitational constant, and define the reduced Planck mass and the Planck mass as, M = (8πG )−1=2 2:4 1018 GeV ; m = (G )−1=2 1:23 1019 GeV : (1) P N ' × P N ' × Cosmic inflation is a period of an accelerated expansion that is believed to have occured in an early epoch of the Universe, roughly speaking at an energy scale, E 1016 GeV, for which the Hubble I ∼ parameter, H E2=M 1013 GeV. I ∼ I P ∼ The dynamics of a homogeneous expanding Universe with the line element, ds2 = dt2 a2d~x 2 (2) − is governed by the FLRW equation, a¨ 4πG Λ = N (ρ + 3 ) + ; (3) a − 3 P 3 from which it follows that an accelerated expansion, a¨ > 0, is realised either when the active gravitational energy density is negative, ρ = ρ + 3 < 0 ; (4) active P or when there is a positive cosmological term, Λ > 0. While we have no idea how to realise Λ in labo- ratory, a negative gravitational mass could be created by a scalar matter with a nonvanishing potential energy. Indeed, from the energy density and pressure in a scalar field in an isotropic background, 1 1 ρ = '_ 2 + ( ')2 + V (') ' 2 2 r 1 1 = '_ 2 + ( ')2 V (') (5) P' 2 6 r − we see that ρ = 2'_ 2 + ( ')2 2V (') ; (6) active r − can be negative, provided the potential energy is greater than twice the kinetic plus gradient energy, V > '_ 2 + ( ')2, = @~=a. -
Orders of Magnitude (Length) - Wikipedia
03/08/2018 Orders of magnitude (length) - Wikipedia Orders of magnitude (length) The following are examples of orders of magnitude for different lengths. Contents Overview Detailed list Subatomic Atomic to cellular Cellular to human scale Human to astronomical scale Astronomical less than 10 yoctometres 10 yoctometres 100 yoctometres 1 zeptometre 10 zeptometres 100 zeptometres 1 attometre 10 attometres 100 attometres 1 femtometre 10 femtometres 100 femtometres 1 picometre 10 picometres 100 picometres 1 nanometre 10 nanometres 100 nanometres 1 micrometre 10 micrometres 100 micrometres 1 millimetre 1 centimetre 1 decimetre Conversions Wavelengths Human-defined scales and structures Nature Astronomical 1 metre Conversions https://en.wikipedia.org/wiki/Orders_of_magnitude_(length) 1/44 03/08/2018 Orders of magnitude (length) - Wikipedia Human-defined scales and structures Sports Nature Astronomical 1 decametre Conversions Human-defined scales and structures Sports Nature Astronomical 1 hectometre Conversions Human-defined scales and structures Sports Nature Astronomical 1 kilometre Conversions Human-defined scales and structures Geographical Astronomical 10 kilometres Conversions Sports Human-defined scales and structures Geographical Astronomical 100 kilometres Conversions Human-defined scales and structures Geographical Astronomical 1 megametre Conversions Human-defined scales and structures Sports Geographical Astronomical 10 megametres Conversions Human-defined scales and structures Geographical Astronomical 100 megametres 1 gigametre -
Distances in Cosmology
Distances in Cosmology REVIEW OF “WORLD MODELS” Simplify notation by adopting c=1, so that E=m. Friedmann's equation is then: . Substitute H(t)= a/a and recall that we can formally associate an energy density with the cosmological constant, i.e The index i refers to the type of particle fluid under consideration, e.g. matter or radiation. If the Universe is flat (k=0): Let us define the fraction of the critical density contributed by each component of the Universe : So that we have Ωm , Ωr and ΩΛ for matter, radiation and dark energy. These quantities are time-dependent, the values today are denoted as Ωm,0 We can rewrite the Friedmann eqn: If we define Ωk = -k/(aH)2, we can write: Flat FRW Cosmologies In the last lecture we showed that the density of matter evolves as: Set a0=1. The Friedmann equation becomes: or In a flat Universe, ΩΛ,0 + Ωm,0=1. Case 1: Λ>0 Use the substitution: to obtain Take the positive root: This can be integrated by completing the square in the u-integral and with substitutions v = u + 1 and cosh w = v: to yield the solution: Case 2, Λ < 0: Introduce Solution: CaseCase 33,, Λ=0 : This is now the Einstein-deSitter case which we have already encountered in the last lecture. A flat, pressureless universe with a small, but non-zero, cosmo- logical constant initially evolves as if it were Einstein-deSitter. Fo r Λ > 0, the second term on the right-hand side of these equations dominates at large values of t and the universe grows exponentially: A wide variety of world models are conceivable, depending on the values of the parameters Ω.