DOCSLIB.ORG

Lecture 5: Building a Model - Dynamics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 5: Building a Model - Dynamics Lecture 5: Building a model - dynamics 1. A Newtonian derivation of the Friedmann equation 2. Newtonian to General Relativity (conceptually) 3. The Friedmann Acceleration equation 4. Einstein’s Blunder? • Adding in the Cosmological constant (Λ) • Equation of State of Λ (w=-1) • What is Λ? 5. The Hubble constant revisited 6. Endpoints of the Universe Course Text: Chapter 4 & 6 Wikipedia: Friedmann Equations, Einstein The Robertson-Walker metric Space-time interval/metric Normal spatial polar coords 2 2 2 2 2 % dr 2 2 2 2 ( ds = c dt " R (t) ' 2 + r (d# + sin #d$ )* & 1" k r ) Light travel distance Curvature Term (open, closed, flat) Expansion factor ! This metric is forced by the Cosmological Principle (universal expansion) Spacetime interval for a photon = 0 (from special relativity) If we consider photons only on radial paths in a flat Universe (k=0): c 2dt 2 = R2(t)dr2 ! The Dynamics of the Universe • We currently have a geometric description of the Universe but we lack a description of the dynamics R(t)=? • Notation: R0 = R(t = T), scale-factor today at time T r = R(t), distance dR(t) r! = , velocity dt d 2 R(t) !r!= , acceleration dt 2 d 3R(t) !r!!= , rate of change of acceleration dt3 • GR tells us dynamics depend on contents (interlude) Einstein’s General Relativity • Spacetime geometry tells matter how to move – gravity = effect of curved spacetime – free particles follow geodessic trajectories – ds2 > 0 v < c time-like massive particles – ds2 = 0 v = c null massless particles (photons) – ds2 < 0 v > c space-like tachyons (not observed) • Matter (+energy) tells spacetime how to curve – Einstein field equations – nonlinear – second-order derivatives of metric with respect to space/time coordinates Spacetime Einstein’s Theory of Relativity • The laws of Physics are the same to all observers, i.e., there is no fundamental reference frame. • The speed-of-light represents a fundamental limit and the velocity of a photon is measured to be the same to all observers • Counter-intuitive but has been tested exhaustively! The Equivalence Principle This states that you cannot distinguish between being at rest in a gravitational field and acceleration. i.e., gravity and inertia are indistinguishable GR equates inertial mass with gravitational mass and explains why they are identical Einstein Field Equations GEOMETRY=CONTENTS Gµ" = 8# G Tµ" + $ gµ" 2 µ " gµ" = spacetime metric ( ds = gµ" dx dx ) Gµ" = Einstein tensor (spacetime curvature) G = Netwon's gravitational constant Tµ" = energy - momentum tensor $ = cosmological constant ! The Friedmann eqns: • Adopt standard GR G = 8#GT + $g • Assume µ" µ" µ" – Homogeneity and Isotropy, i.e., uniform fluid – Adiabatic expansion (lossless!) "E = #p"V ! Hubble constant or H(t) > Friedmann eqns: 2 ! • $ R 8!G" kc2 ( Λ = Cosmological constant # & = ' 2 + = vacuum energy ! # R & 3 R 3 = dark energy " % = “Greatest blunder” •• = Inflation R 4!G ! 3p$ ( = ' #" + &+ = Useful Fudge factor? R 3 " c2 % 3 Dynamics of the Universe • What contents do we know about? -3 – Matter, ρm~r -4 – Radiation, ρr~r – Intrinsic curvature/quantum froth/cosmological constant, ρv~ k – Anything else? • Adopting CP allows us to describe the Universe as a smooth fluid comprised of radiation and matter and any intrinsic space-time curvature • Adoption of adiabatic (lossless expansion): dE=-pdV – Fluid equation: • • r p "+ 3 (" + ) = 0 r c 2 • So we have an expression involving change in densities with velocity of the scale factor so we only need an expression for the acceleration/deceleration! (how this changes with time). From Lec 1: Adiabatic Expansion • If U self-contained it must expand without losing energy: (1st law of thermodynamics) dE = "pdV • Can use E=mc2 and rewrite with m= ρ(4πr3/3) where r is some physical radius for expanding region of density ρ. dE d(!4" r3#c 2 3) 4 d# dr dr = = "r3c 2 + #4"r2c 2 = $4"pr2 = $pdV dt dt 3 dt dt dt [Uses: Chain rule + d(x3)=3x2d ! dx • 4 • • • • Use dot notation: i.e., = x "r3c 2 #+ 4"r2c 2# r = $4"r2 pr 3 dt • • Rearrange to get the Fluid Equation: • r p "+ 3 (" + ) = 0 r c 2 ! ! ! Newtonian Analogy Consider, a location and a test particle: CP tells us: • Any point can be the centre • Any location the particle m Particle only feels forces from inside sphere as d gravity from shells outside R acting on m cancel (Newton’s Theorem) " Total energy given by U=T+V (potential+kinetic) 2 1 2 GMm 1 2 4!G"d m U = md! ! = md! ! 2 d 2 3 ! d = R(t)r1 = rr1 2 2 2 2 2 1 "d(rr )% 4!G"r r m 1 2 2 4!G"r r m U = 1 ! 1 = r! r ! 1 2#$ dt &' 3 2 1 3 Rearrange : 2 ( r!+ 8!G" 2U * - = + 2 2 ) r , 3 mr1 r Newton à GR From analogy to GR form: 2 2U Set: kc = ! 2 mr1 2 " r!% 8!G" kc2 $ ' = ! # r & 3 r2 !2U [k = 2 2 = curvature term in RW metric=0 if Universe flat] mc r1 Now differentiate this expression using the Fluid Eqn: 2 1 d r! d( 2 ) 8!G!! 2 ( r ) = ! kc r dt 3 dt LHS: 2 1 1 d(r! 2 ) 2 d( 2 ) 2 2 1 d(r! ) 2 2r!!r! 2r! r! r! r!r!! r! r = + r! r = ! = 2 [ ] dt r2 dt dt r2 r3 r r2 1 2 d( 2 ) r! r!r!! r! 8!G!! 2 2 [ ] = ! kc r r r2 3 dt Sub for ! from fluid equation: r! r!r!! r!2 8!G r!" p % 2kc2r! 2 [ ] = ! 3 $! + '+ r r2 3 r # c2 & r3 !r! r!2 " p % kc2 ! = !4!G$" + '+ r r2 # c2 & r2 Sub for: (r! )2 and rearrange to get: r !r! p 8!G" = !4!G" ! 4!G + r c2 3 !r! 4!G 3p = ! (" + ) r 3 c2 Deceleration equation Basic Friedmann Equations Curvature term k=0=FLAT 2 2 k=-1=OPEN Velocity of ! r!$ 8!G" kc k=+1=CLOSED Scale factor # & = ' " r % 3 r2 !r! 4!G 3p Deceleration of = ' (" + ) Scale factor r 3 c2 Contents acting via density and pressure Problem: Unless universe is empty it will decelerate, i.e., non-static We measure both matter and radiation to be positive. Einstein wanted a static (infinite) Universe Einstein’s Blunder? • Einstein introduced a constant density, ρv • How does constant density behave? dE = !pdV d(!R3 ) = !p / c2d(R3 ) If !v = constant p ! = ! v c2 Exerts -ve pressure!!?! 2 (NB : in equation of state: w!vc = p, so w=-1) • Physical motivations for Λ? – Quantum froth – particles flicking in and out of existence – Intrinsic curvature – intrinsic curvature of empty spacetime ? – Dark Energy – hypothetical energy which pervades spacetime Λ 2 ! r!$ 8!G(" + " + " ) kc2 # & = M R v ' " r % 3 r2 2 ! r!$ 8!G" kc2 8!G" # & = ' + v " r % 3 r2 3 Define: ( = 8!G"v 2 2 ! r!$ 8!G" kc ( The Friedmann # & = ' + Equation " r % 3 r2 3 Λ introduced to create a static universe by providing an acceleration to counter-balance the deceleration implied by matter and radiation. Λ !r! 4!G 3(p + p + p ) = ! (" + " + " + M R v ) r 3 M R v c2 2 But : pv = !"vc => !r! 4!G 3p = ! (" + ! 2" ) r 3 c2 v !r! 4!G 3p 8!G = ! (" + )+ " r 3 c2 3 v !r! 4!G 3p " = ! (" + )+ Friedmann Acceleration r 3 c2 3 Equation Λ introduced to create a static universe by providing an acceleration to counter-balance the deceleration implied by matter and radiation. Notation • Todays values carry a suffix, i.e., R(t) and R0 – i.e., ρ à ρo etc The Hubble Constant v ! r!$ ! r!$ • Hubble constant is: H0 = = # & , H(t)=# & d " r % " r % 0 • Hubble constant is only constant in space not time • Can rewrite Friedmann equation now with H: 2 2 8!G" kc " H = ! + 3 r2 3 Question? • Show for a flat Universe that the critical density can be given by: 3H 2 ! = 0 crit 8"G • In a flat Universe k=0, assuming Λ=0 2 ! r!$ 8!G" kc2 ( # & = ' + " r % 3 r2 3 2 8!G" H = crit o 3 3H 2 " = o ) 10'26 kg / m3 crit 8!G If "o > "crit k is +ve Universe is closed (will recollapse) If "o < "crit k is -ve Universe is open (will expand forever) " kc2 [i.e., 1' o = ' ] H 2r2 "crit 0 Expand forever or recollapse? Empty Open Flat 1/H0 Closed Size of Universeof Size Now Time With Λ Did we actually need GR? 2 ! r!$ 8!G" 2U ' # & = + 2 2 + Newtonian " r % 3 mr1 r 3 2 ! r!$ 8!G" kc2 ' # & = ( + GR " r % 3 r2 3 For a static Universe U=0, i.e., potential and kinetic energies are balanced. For a static flat Universe Newton = GR For any non-static Universe the extra term manifests as an energy imbalance in the Newtonian Universe but a curvature in the GR Universe which are fundamentally different. GR’s success at explaining precession of Mercury, occultation times, and gravitational lensing favours it but Newtonian models are still being explored. Conclusion: GR is required if either motion or curvature exist. =>Expansion demands GR or Newton interpretation. .
Load more

Recommended publications
  • Galaxies 2013, 1, 1-5; Doi:10.3390/Galaxies1010001
    Galaxies 2013, 1, 1-5; doi:10.3390/galaxies1010001 OPEN ACCESS galaxies ISSN 2075-4434 www.mdpi.com/journal/galaxies Editorial Galaxies: An International Multidisciplinary Open Access Journal Emilio Elizalde Consejo Superior de Investigaciones Científicas, Instituto de Ciencias del Espacio & Institut d’Estudis Espacials de Catalunya, Faculty of Sciences, Campus Universitat Autònoma de Barcelona, Torre C5-Parell-2a planta, Bellaterra (Barcelona) 08193, Spain; E-Mail: [email protected]; Tel.: +34-93-581-4355 Received: 23 October 2012 / Accepted: 1 November 2012 / Published: 8 November 2012 The knowledge of the universe as a whole, its origin, size and shape, its evolution and future, has always intrigued the human mind. Galileo wrote: “Nature’s great book is written in mathematical language.” This new journal will be devoted to both aspects of knowledge: the direct investigation of our universe and its deeper understanding, from fundamental laws of nature which are translated into mathematical equations, as Galileo and Newton—to name just two representatives of a plethora of past and present researchers—already showed us how to do. Those physical laws, when brought to their most extreme consequences—to their limits in their respective domains of applicability—are even able to give us a plausible idea of how the origin of our universe came about and also of how we can expect its future to evolve and, eventually, how its end will take place. These laws also condense the important interplay between mathematics and physics as just one first example of the interdisciplinarity that will be promoted in the Galaxies Journal. Although already predicted by the great philosopher Immanuel Kant and others before him, galaxies and the existence of an “Island Universe” were only discovered a mere century ago, a fact too often forgotten nowadays when we deal with multiverses and the like.
    [Show full text]
  • Some Aspects in Cosmological Perturbation Theory and F (R) Gravity
    Some Aspects in Cosmological Perturbation Theory and f (R) Gravity Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn von Leonardo Castañeda C aus Tabio,Cundinamarca,Kolumbien Bonn, 2016 Dieser Forschungsbericht wurde als Dissertation von der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Bonn angenommen und ist auf dem Hochschulschriftenserver der ULB Bonn http://hss.ulb.uni-bonn.de/diss_online elektronisch publiziert. 1. Gutachter: Prof. Dr. Peter Schneider 2. Gutachter: Prof. Dr. Cristiano Porciani Tag der Promotion: 31.08.2016 Erscheinungsjahr: 2016 In memoriam: My father Ruperto and my sister Cecilia Abstract General Relativity, the currently accepted theory of gravity, has not been thoroughly tested on very large scales. Therefore, alternative or extended models provide a viable alternative to Einstein’s theory. In this thesis I present the results of my research projects together with the Grupo de Gravitación y Cosmología at Universidad Nacional de Colombia; such projects were motivated by my time at Bonn University. In the first part, we address the topics related with the metric f (R) gravity, including the study of the boundary term for the action in this theory. The Geodesic Deviation Equation (GDE) in metric f (R) gravity is also studied. Finally, the results are applied to the Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime metric and some perspectives on use the of GDE as a cosmological tool are com- mented. The second part discusses a proposal of using second order cosmological perturbation theory to explore the evolution of cosmic magnetic fields. The main result is a dynamo-like cosmological equation for the evolution of the magnetic fields.
    [Show full text]
  • Open Sloan-Dissertation.Pdf
    The Pennsylvania State University The Graduate School LOOP QUANTUM COSMOLOGY AND THE EARLY UNIVERSE A Dissertation in Physics by David Sloan c 2010 David Sloan Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2010 The thesis of David Sloan was reviewed and approved∗ by the following: Abhay Ashtekar Eberly Professor of Physics Dissertation Advisor, Chair of Committee Martin Bojowald Professor of Physics Tyce De Young Professor of Physics Nigel Higson Professor of Mathematics Jayanth Banavar Professor of Physics Head of the Department of Physics ∗Signatures are on file in the Graduate School. Abstract In this dissertation we explore two issues relating to Loop Quantum Cosmology (LQC) and the early universe. The first is expressing the Belinkskii, Khalatnikov and Lifshitz (BKL) conjecture in a manner suitable for loop quantization. The BKL conjecture says that on approach to space-like singularities in general rela- tivity, time derivatives dominate over spatial derivatives so that the dynamics at any spatial point is well captured by a set of coupled ordinary differential equa- tions. A large body of numerical and analytical evidence has accumulated in favor of these ideas, mostly using a framework adapted to the partial differential equa- tions that result from analyzing Einstein's equations. By contrast we begin with a Hamiltonian framework so that we can provide a formulation of this conjecture in terms of variables that are tailored to non-perturbative quantization. We explore this system in some detail, establishing the role of `Mixmaster' dynamics and the nature of the resulting singularity. Our formulation serves as a first step in the analysis of the fate of generic space-like singularities in loop quantum gravity.
    [Show full text]
  • The Cosmological Constant Einstein’S Static Universe
    Some History TEXTBOOKS FOR FURTHER REFERENCE 1) Physical Foundations of Cosmology, Viatcheslav Mukhanov 2) Cosmology, Michael Rowan-Robinson 3) A short course in General Relativity, J. Foster and J.D. Nightingale DERIVATION OF FRIEDMANN EQUATIONS IN A NEWTONIAN COSMOLOGY THE COSMOLOGICAL PRINCIPAL Viewed on a sufficiently large scale, the properties of the Universe are the same for all observers. This amounts to the strongly philosophical statement that the part of the Universe which we can see is a fair sample, and that the same physical laws apply throughout. => the Hubble expansion is a natural property of an expanding universe which obeys the cosmological principle Distribution of galaxies on the sky Distribution of 2.7 K microwave radiation on the sky vA = H0. rA vB = H0. rB v and r are position and velocity vectors VBA = VB – VA = H0.rB – H0.rA = H0 (rB - rA) The observer on galaxy A sees all other galaxies in the universe receding with velocities described by the same Hubble law as on Earth. In fact, the Hubble law is the unique expansion law compatible with homogeneity and isotropy. Co-moving coordinates: express the distance r as a product of the co-moving distance x and a term a(t) which is a function of time only: rBA = a(t) . x BA The original r coordinate system is known as physical coordinates. Deriving an equation for the universal expansion thus reduces to determining a function which describes a(t) Newton's Shell Theorem The force acting on A, B, C, D— which are particles located on the surface of a sphere of radius r—is the gravitational attraction from the matter internal to r only, acting as a point mass at O.
    [Show full text]
  • Formation of Structure in Dark Energy Cosmologies
    HELSINKI INSTITUTE OF PHYSICS INTERNAL REPORT SERIES HIP-2006-08 Formation of Structure in Dark Energy Cosmologies Tomi Sebastian Koivisto Helsinki Institute of Physics, and Division of Theoretical Physics, Department of Physical Sciences Faculty of Science University of Helsinki P.O. Box 64, FIN-00014 University of Helsinki Finland ACADEMIC DISSERTATION To be presented for public criticism, with the permission of the Faculty of Science of the University of Helsinki, in Auditorium CK112 at Exactum, Gustaf H¨allstr¨omin katu 2, on November 17, 2006, at 2 p.m.. Helsinki 2006 ISBN 952-10-2360-9 (printed version) ISSN 1455-0563 Helsinki 2006 Yliopistopaino ISBN 952-10-2961-7 (pdf version) http://ethesis.helsinki.fi Helsinki 2006 Helsingin yliopiston verkkojulkaisut Contents Abstract vii Acknowledgements viii List of publications ix 1 Introduction 1 1.1Darkenergy:observationsandtheories..................... 1 1.2Structureandcontentsofthethesis...................... 6 2Gravity 8 2.1Generalrelativisticdescriptionoftheuniverse................. 8 2.2Extensionsofgeneralrelativity......................... 10 2.2.1 Conformalframes............................ 13 2.3ThePalatinivariation.............................. 15 2.3.1 Noethervariationoftheaction..................... 17 2.3.2 Conformalandgeodesicstructure.................... 18 3 Cosmology 21 3.1Thecontentsoftheuniverse........................... 21 3.1.1 Darkmatter............................... 22 3.1.2 Thecosmologicalconstant........................ 23 3.2Alternativeexplanations............................
    [Show full text]
  • The Dark Energy of the Universe the Dark Energy of the Universe
    The Dark Energy of the Universe The Dark Energy of the Universe Jim Cline, McGill University Niels Bohr Institute, 13 Oct., 2015 image: bornscientist.com J.Cline, McGill U. – p. 1 Dark Energy in a nutshell In 1998, astronomers presented evidence that the primary energy density of the universe is not from particles or radiation, but of empty space—the vacuum. Einstein had predicted it 80 years earlier, but few people believed this prediction, not even Einstein himself. Many scientists were surprised, and the discovery was considered revolutionary. Since then, thousands of papers have been written on the subject, many speculating on the detailed properties of the dark energy. The fundamental origin of dark energy is the subject of intense controversy and debate amongst theorists. J.Cline, McGill U. – p. 2 Outline Part I History of the dark energy • Theory of cosmological expansion • The observational evidence for dark energy • Part II What could it be? • Upcoming observations • The theoretical crisis !!! • J.Cline, McGill U. – p. 3 Albert Einstein invents dark energy, 1917 Two years after introducing general relativity (1915), Einstein looks for cosmological solutions of his equations. No static solution exists, contrary to observed universe at that time He adds new term to his equations to allow for static universe, the cosmological constant λ: J.Cline, McGill U. – p. 4 Einstein’s static universe This universe is a three-sphere with radius R and uniform mass density of stars ρ (mass per volume). mechanical analogy potential energy R R By demanding special relationships between λ, ρ and R, λ = κρ/2 = 1/R2, a static solution can be found.
    [Show full text]
  • 3.1 the Robertson-Walker Metric
    M. Pettini: Introduction to Cosmology | Lecture 3 RELATIVISTIC COSMOLOGY 3.1 The Robertson-Walker Metric The appearance of objects at cosmological distances is affected by the curvature of spacetime through which light travels on its way to Earth. The most complete description of the geometrical properties of the Universe is provided by Einstein's general theory of relativity. In GR, the fundamental quantity is the metric which describes the geometry of spacetime. Let's look at the definition of a metric: in 3-D space we measure the distance along a curved path P between two points using the differential distance formula, or metric: (d`)2 = (dx)2 + (dy)2 + (dz)2 (3.1) and integrating along the path P (a line integral) to calculate the total distance: Z 2 q Z 2 q ∆` = (d`)2 = (dx)2 + (dy)2 + (dz)2 (3.2) 1 1 Similarly, to measure the interval along a curved wordline, W, connecting two events in spacetime with no mass present, we use the metric for flat spacetime: (ds)2 = (cdt)2 − (dx)2 − (dy)2 − (dz)2 (3.3) Integrating ds gives the total interval along the worldline W: Z B q Z B q ∆s = (ds)2 = (cdt)2 − (dx)2 − (dy)2 − (dz)2 (3.4) A A By definition, the distance measured between two events, A and B, in a reference frame for which they occur simultaneously (tA = tB) is the proper distance: q ∆L = −(∆s)2 (3.5) Our search for a metric that describes the spacetime of a matter-filled universe, is made easier by the cosmological principle.
    [Show full text]
  • Universes with a Cosmological Constant
    Universes with a cosmological constant Domingos Soares Physics Department Federal University of Minas Gerais Belo Horizonte, MG, Brasil July 21, 2015 Abstract I present some relativistic models of the universe that have the cos- mological constant (Λ) in their formulation. Einstein derived the first of them, inaugurating the theoretical strand of the application of the field equations of General Relativity with the cosmological constant. One of the models shown is the Standard Model of Cosmology, which presently enjoys the support of a significant share of the scientific community. 1 Introduction The first cosmological model based in General Relativity Theory (GRT) was put forward in 1917 by the creator of GRT himself. Einstein conceived the universe as a static structure and, to obtain the corresponding relativistic model, introduced a repulsive component in the formulation of the field equations of GRT to avoid the collapse produced by the matter. Such a component appears in the equations as an additional term of the metric field multiplied by a constant, the so-called cosmological constant (see [1, eq. 10]). The cosmological constant is commonly represented by the uppercase Greek letter (Λ) and has physical dimensions of 1/length2. Besides Einstein's static model, other relativistic models with the cos- mological constant were proposed. We will see that these models can be obtained from Friedmann's equation plus the cosmological constant, and by 1 means of the appropriate choices of the spatial curvature and of the matter- energy content of the universe. The cosmologist Steven Weinberg presents several of those models in his book Gravitation and Cosmology, in the chap- ter entitled Models with a Cosmological Constant [2, p.
    [Show full text]
  • The Discovery of the Expansion of the Universe
    galaxies Review The Discovery of the Expansion of the Universe Øyvind Grøn Faculty of Technology, Art and Design, Oslo Metropolitan University, PO Box 4 St. Olavs Plass, NO-0130 Oslo, Norway; [email protected]; Tel.: +047-90-94-64-60 Received: 2 November 2018; Accepted: 29 November 2018; Published: 3 December 2018 Abstract: Alexander Friedmann, Carl Wilhelm Wirtz, Vesto Slipher, Knut E. Lundmark, Willem de Sitter, Georges H. Lemaître, and Edwin Hubble all contributed to the discovery of the expansion of the universe. If only two persons are to be ranked as the most important ones for the general acceptance of the expansion of the universe, the historical evidence points at Lemaître and Hubble, and the proper answer to the question, “Who discovered the expansion of the universe?”, is Georges H. Lemaître. Keywords: cosmology history; expansion of the universe; Lemaitre; Hubble 1. Introduction The history of the discovery of the expansion of the universe is fascinating, and it has been thoroughly studied by several historians of science. (See, among others, the contributions to the conference Origins of the expanding universe [1]: 1912–1932). Here, I will present the main points of this important part of the history of the evolution of the modern picture of our world. 2. Einstein’s Static Universe Albert Einstein completed the general theory of relativity in December 1915, and the theory was presented in an impressive article [2] in May 1916. He applied [3] the theory to the construction of a relativistic model of the universe in 1917. At that time, it was commonly thought that the universe was static, since one had not observed any large scale motions of the stars.
    [Show full text]
  • 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 .
    [Show full text]
  • On Using the Cosmic Microwave Background Shift Parameter in Tests of Models of Dark Energy
    A&A 471, 65–70 (2007) Astronomy DOI: 10.1051/0004-6361:20077292 & c ESO 2007 Astrophysics On using the cosmic microwave background shift parameter in tests of models of dark energy Ø. Elgarøy1 and T. Multamäki2 1 Institute of theoretical astrophysics, University of Oslo, Box 1029, 0315 Oslo, Norway e-mail: [email protected] 2 Department of Physics, University of Turku, 20014 Turku, Finland e-mail: [email protected] Received 13 February 2007 / Accepted 31 May 2007 ABSTRACT Context. The so-called shift parameter is related to the position of the first acoustic peak in the power spectrum of the temperature anisotropies of the cosmic microwave background (CMB). It is an often used quantity in simple tests of dark energy models. However, the shift parameter is not directly measurable from the cosmic microwave background, and its value is usually derived from the data assuming a spatially flat cosmology with dark matter and a cosmological constant. Aims. To evaluate the effectiveness of the shift parameter as a constraint on dark energy models. Methods. We discuss the potential pitfalls in using the shift parameter as a test of non-standard dark energy models. Results. By comparing to full CMB fits, we show that combining the shift parameter with the position of the first acoustic peak in the CMB power spectrum improves the accuracy of the test considerably. Key words. cosmology: theory – cosmological parameters 1. Introduction understanding of dark energy argues for keeping an open mind. One would like to have some means of testing both groups of Comparing cosmological models to current observational data models, incorporating as much empirical information as pos- can be cumbersome and computationally intensive.
    [Show full text]
  • FRW (Friedmann-Robertson-Walker) Universe
    Chapter 2 FRW (Friedmann-Robertson-Walker) universe We start our discussion from the evolution of the homogeneous and isotropic expanding universe, of- ten called Friedmann-Robertson-Walker (FRW) world model. This somewhat idealized world model is consistent with the observations thus far: Hubble diagrams (which shows the distance-redshift relation) have been continuously updating to confirm that the Universe is expanding, observed cosmic microwave background radiation is isotropic from all lines of sight, and distribution of galaxies in SDSS appears to be homogeneous on scales larger than 70 Mpc=h. ' In this chapter, we study the kinematics and dynamics of the FRW universe. 2.1 FRW metric The spacetime of spatially homogeneous and isotropic, expanding universe can be best thought of as a contiguous bundle of homogeneous and isotropic spatial (constant-time) hypersurfaces whose length scale evolves in time. The metric in such a spacetime can be written as1 2 µ ν 2 2 i j ds = gµνd x d x = d t + a (t)g˜i j(x)d x d x , (2.1) − where g˜i j(x) is a time-independent spatial metric defined on the constant-time hypersurface, and a(t) is the scale factor that relates the proper (physical) distance to the comoving (coordinate) distance. As we shall show below, except for the three maximally symmetric cases2—Minkowski, de-Sitter, and anti de-Sitter spacetime, defining the space-like hypersurface can be naturally done with, e.g. constant- density-hypersurface, in the FRW world model. A word about the convention. Throughout the course, we shall use the Greek for the spacetime in- dices µ, ν, = 0, 1, 2, 3 with the index 0 being the time index, and the Latin for the space indices ··· 1 The following form of metric is uniquely dictated from spatial homogeneity and isotropy as we should not have any g0i component which would introduce a preferred direction nor the space dependence in g00 as that breaks homogeneity.
    [Show full text]