General Relativity and Cosmology: Unsolved Questions and Future Directions
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Frame Covariance and Fine Tuning in Inflationary Cosmology
FRAME COVARIANCE AND FINE TUNING IN INFLATIONARY COSMOLOGY A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2019 By Sotirios Karamitsos School of Physics and Astronomy Contents Abstract 8 Declaration 9 Copyright Statement 10 Acknowledgements 11 1 Introduction 13 1.1 Frames in Cosmology: A Historical Overview . 13 1.2 Modern Cosmology: Frames and Fine Tuning . 15 1.3 Outline . 17 2 Standard Cosmology and the Inflationary Paradigm 20 2.1 General Relativity . 20 2.2 The Hot Big Bang Model . 25 2.2.1 The Expanding Universe . 26 2.2.2 The Friedmann Equations . 29 2.2.3 Horizons and Distances in Cosmology . 33 2.3 Problems in Standard Cosmology . 34 2.3.1 The Flatness Problem . 35 2.3.2 The Horizon Problem . 36 2 2.4 An Accelerating Universe . 37 2.5 Inflation: More Questions Than Answers? . 40 2.5.1 The Frame Problem . 41 2.5.2 Fine Tuning and Initial Conditions . 45 3 Classical Frame Covariance 48 3.1 Conformal and Weyl Transformations . 48 3.2 Conformal Transformations and Unit Changes . 51 3.3 Frames in Multifield Scalar-Tensor Theories . 55 3.4 Dynamics of Multifield Inflation . 63 4 Quantum Perturbations in Field Space 70 4.1 Gauge Invariant Perturbations . 71 4.2 The Field Space in Multifield Inflation . 74 4.3 Frame-Covariant Observable Quantities . 78 4.3.1 The Potential Slow-Roll Hierarchy . 81 4.3.2 Isocurvature Effects in Two-Field Models . 83 5 Fine Tuning in Inflation 88 5.1 Initial Conditions Fine Tuning . -
The Planck Satellite and the Cosmic Microwave Background
The Cosmic Microwave Background, Dark Matter and Dark Energy Anthony Lasenby, Astrophysics Group, Cavendish Laboratory and Kavli Institute for Cosmology, Cambridge Overview The Cosmic Microwave Background — exciting new results from the Planck Satellite Context of the CMB =) addressing key questions about the Big Bang and the Universe, including Dark Matter and Dark Energy Planck Satellite and planning for its observations have been a long time in preparation — first meetings in 1993! UK has been intimately involved Two instruments — the LFI (Low — e.g. Cambridge is the Frequency Instrument) and the HFI scientific data processing (High Frequency Instrument) centre for the HFI — RAL provided the 4K Cooler The Cosmic Microwave Background (CMB) So what is the CMB? Anywhere in empty space at the moment there is radiation present corresponding to what a blackbody would emit at a temperature of ∼ 2:74 K (‘Blackbody’ being a perfect emitter/absorber — furnace with a small opening is a good example - needs perfect thermodynamic equilibrium) CMB spectrum is incredibly accurately black body — best known in nature! COBE result on this showed CMB better than its own reference b.b. within about 9 minutes of data! Universe History Radiation was emitted in the early universe (hot, dense conditions) Hot means matter was ionised Therefore photons scattered frequently off the free electrons As universe expands it cools — eventually not enough energy to keep the protons and electrons apart — they History of the Universe: superluminal inflation, particle plasma, -
PDF Solutions
Solutions to exercises Solutions to exercises Exercise 1.1 A‘stationary’ particle in anylaboratory on theEarth is actually subject to gravitationalforcesdue to theEarth andthe Sun. Thesehelp to ensure that theparticle moveswith thelaboratory.Ifstepsweretaken to counterbalance theseforcessothatthe particle wasreally not subject to anynet force, then the rotation of theEarth andthe Earth’sorbital motionaround theSun would carry thelaboratory away from theparticle, causing theforce-free particle to followacurving path through thelaboratory.Thiswouldclearly show that the particle didnot have constantvelocity in the laboratory (i.e.constantspeed in a fixed direction) andhence that aframe fixed in the laboratory is not an inertial frame.More realistically,anexperimentperformed usingthe kind of long, freely suspendedpendulum known as a Foucaultpendulum couldreveal the fact that a frame fixed on theEarth is rotating andthereforecannot be an inertial frame of reference. An even more practical demonstrationisprovidedbythe winds,which do not flowdirectly from areas of high pressure to areas of lowpressure because of theEarth’srotation. - Exercise 1.2 TheLorentzfactor is γ(V )=1/ 1−V2/c2. (a) If V =0.1c,then 1 γ = - =1.01 (to 3s.f.). 1 − (0.1c)2/c2 (b) If V =0.9c,then 1 γ = - =2.29 (to 3s.f.). 1 − (0.9c)2/c2 Notethatitisoften convenient to write speedsinterms of c instead of writingthe values in ms−1,because of thecancellation between factorsofc. ? @ AB Exercise 1.3 2 × 2 M = Theinverse of a matrix CDis ? @ 1 D −B M −1 = AD − BC −CA. Taking A = γ(V ), B = −γ(V )V/c, C = −γ(V)V/c and D = γ(V ),and noting that AD − BC =[γ(V)]2(1 − V 2/c2)=1,wehave ? @ γ(V )+γ(V)V/c [Λ]−1 = . -
Kaluza-Klein Gravity, Concentrating on the General Rel- Ativity, Rather Than Particle Physics Side of the Subject
Kaluza-Klein Gravity J. M. Overduin Department of Physics and Astronomy, University of Victoria, P.O. Box 3055, Victoria, British Columbia, Canada, V8W 3P6 and P. S. Wesson Department of Physics, University of Waterloo, Ontario, Canada N2L 3G1 and Gravity Probe-B, Hansen Physics Laboratories, Stanford University, Stanford, California, U.S.A. 94305 Abstract We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identi- fied and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time. arXiv:gr-qc/9805018v1 7 May 1998 Preprint submitted to Elsevier Preprint 3 February 2008 1 Introduction Kaluza’s [1] achievement was to show that five-dimensional general relativity contains both Einstein’s four-dimensional theory of gravity and Maxwell’s the- ory of electromagnetism. He however imposed a somewhat artificial restriction (the cylinder condition) on the coordinates, essentially barring the fifth one a priori from making a direct appearance in the laws of physics. Klein’s [2] con- tribution was to make this restriction less artificial by suggesting a plausible physical basis for it in compactification of the fifth dimension. This idea was enthusiastically received by unified-field theorists, and when the time came to include the strong and weak forces by extending Kaluza’s mechanism to higher dimensions, it was assumed that these too would be compact. This line of thinking has led through eleven-dimensional supergravity theories in the 1980s to the current favorite contenders for a possible “theory of everything,” ten-dimensional superstrings. -
Summary As Preparation for the Final Exam
UUnnddeerrsstatannddiinngg ththee UUnniivveerrssee Summary Lecture Volker Beckmann Joint Center for Astrophysics, University of Maryland, Baltimore County & NASA Goddard Space Flight Center, Astrophysics Science Division UMBC, December 12, 2006 OOvveerrvviieeww ● What did we learn? ● Dark night sky ● distances in the Universe ● Hubble relation ● Curved space ● Newton vs. Einstein : General Relativity ● metrics solving the EFE: Minkowski and Robertson-Walker metric ● Solving the Einstein Field Equation for an expanding/contracting Universe: the Friedmann equation Graphic: ESA / V. Beckmann OOvveerrvviieeww ● Friedmann equation ● Fluid equation ● Acceleration equation ● Equation of state ● Evolution in a single/multiple component Universe ● Cosmic microwave background ● Nucleosynthesis ● Inflation Graphic: ESA / V. Beckmann Graphic by Michael C. Wang (UCSD) The velocity- distance relation for galaxies found by Edwin Hubble. Graphic: Edwin Hubble (1929) Expansion in a steady state Universe Expansion in a non-steady-state Universe The effect of curvature The equivalent principle: You cannot distinguish whether you are in an accelerated system or in a gravitational field NNeewwtotonn vvss.. EEiinnssteteiinn Newton: - mass tells gravity how to exert a force, force tells mass how to accelerate (F = m a) Einstein: - mass-energy (E=mc²) tells space time how to curve, curved space-time tells mass-energy how to move (John Wheeler) The effect of curvature A glimpse at EinsteinThe effect’s field of equationcurvature Left side (describes the action -
Inflation, Large Branes, and the Shape of Space
Inflation, Large Branes, and the Shape of Space Brett McInnes National University of Singapore email: [email protected] ABSTRACT Linde has recently argued that compact flat or negatively curved spatial sections should, in many circumstances, be considered typical in Inflationary cosmologies. We suggest that the “large brane instability” of Seiberg and Witten eliminates the negative candidates in the context of string theory. That leaves the flat, compact, three-dimensional manifolds — Conway’s platycosms. We show that deep theorems of Schoen, Yau, Gromov and Lawson imply that, even in this case, Seiberg-Witten instability can be avoided only with difficulty. Using a specific cosmological model of the Maldacena-Maoz type, we explain how to do this, and we also show how the list of platycosmic candidates can be reduced to three. This leads to an extension of the basic idea: the conformal compactification of the entire Euclidean spacetime also has the topology of a flat, compact, four-dimensional space. arXiv:hep-th/0410115v2 19 Oct 2004 1. Nearly Flat or Really Flat? Linde has recently argued [1] that, at least in some circumstances, we should regard cosmological models with flat or negatively curved compact spatial sections as the norm from an Inflationary point of view. Here we wish to argue that cosmic holography, in the novel form proposed by Maldacena and Maoz [2], gives a deep new interpretation of this idea, and also sharpens it very considerably to exclude the negative case. This focuses our attention on cosmological models with flat, compact spatial sections. Current observations [3] show that the spatial sections of our Universe [as defined by observers for whom local isotropy obtains] are fairly close to being flat: the total density parameter Ω satisfies Ω = 1.02 0.02 at 95% confidence level, if we allow the imposition ± of a reasonable prior [4] on the Hubble parameter. -
A Propensity for Genius: That Something Special About Fritz Zwicky (1898 - 1974)
Swiss American Historical Society Review Volume 42 Number 1 Article 2 2-2006 A Propensity for Genius: That Something Special About Fritz Zwicky (1898 - 1974) John Charles Mannone Follow this and additional works at: https://scholarsarchive.byu.edu/sahs_review Part of the European History Commons, and the European Languages and Societies Commons Recommended Citation Mannone, John Charles (2006) "A Propensity for Genius: That Something Special About Fritz Zwicky (1898 - 1974)," Swiss American Historical Society Review: Vol. 42 : No. 1 , Article 2. Available at: https://scholarsarchive.byu.edu/sahs_review/vol42/iss1/2 This Article is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Swiss American Historical Society Review by an authorized editor of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. Mannone: A Propensity for Genius A Propensity for Genius: That Something Special About Fritz Zwicky (1898 - 1974) by John Charles Mannone Preface It is difficult to write just a few words about a man who was so great. It is even more difficult to try to capture the nuances of his character, including his propensity for genius as well as his eccentric behavior edging the abrasive as much as the funny, the scope of his contributions, the size of his heart, and the impact on society that the distinguished physicist, Fritz Zwicky (1898- 1974), has made. So I am not going to try to serve that injustice, rather I will construct a collage, which are cameos of his life and accomplishments. In this way, you, the reader, will hopefully be left with a sense of his greatness and a desire to learn more about him. -
What Is Gravity Probe B? a Quest for Experimental Truth the GP-B Flight
What is Gravity Probe B? than Gravity Probe B. It’s just a star, a telescope, and a spinning sphere.” However, it took the exceptional collaboration of Stanford, Gravity Probe B (GP-B) is a NASA physics mission to experimentally MSFC, Lockheed Martin and a host of others more than four decades investigate Einstein’s 1916 general theory of relativity—his theory of gravity. to develop the ultra-precise gyroscopes and the other cutting- GP-B uses four spherical gyroscopes and a telescope, housed in a satellite edge technologies necessary to carry out this “simple” experiment. orbiting 642 km (400 mi) above the Earth, to measure, with unprecedented accuracy, two extraordinary effects predicted by the general theory of rela- The GP-B Flight Mission & Data Analysis tivity: 1) the geodetic effect—the amount by which the Earth warps the local spacetime in which it resides; and 2) the frame-dragging effect—the amount On April 20, 2004 at 9:57:24 AM PDT, a crowd of over 2,000 current and by which the rotating Earth drags its local spacetime around with it. GP-B former GP-B team members and supporters watched and cheered as the tests these two effects by precisely measuring the precession (displacement) GP-B spacecraft lifted off from Vandenberg Air Force Base. That emotionally angles of the spin axes of the four gyros over the course of a year and com- overwhelming day, culminating with the extraordinary live video of paring these experimental results with predictions from Einstein’s theory. the spacecraft separating from the second stage booster meant, as GP-B Program Manager Gaylord Green put it, “that 10,000 things went right.” A Quest for Experimental Truth Once in orbit, the spacecraft first underwent a four-month Initialization The idea of testing general relativity with orbiting gyroscopes was sug- and Orbit Checkout (IOC), in which all systems and instruments were gested independently by two physicists, George Pugh and Leonard Schiff, initialized, tested, and optimized for the data collection to follow. -
An Analysis of Gravitational Redshift from Rotating Body
An analysis of gravitational redshift from rotating body Anuj Kumar Dubey∗ and A K Sen† Department of Physics, Assam University, Silchar-788011, Assam, India. (Dated: July 29, 2021) Gravitational redshift is generally calculated without considering the rotation of a body. Neglect- ing the rotation, the geometry of space time can be described by using the spherically symmetric Schwarzschild geometry. Rotation has great effect on general relativity, which gives new challenges on gravitational redshift. When rotation is taken into consideration spherical symmetry is lost and off diagonal terms appear in the metric. The geometry of space time can be then described by using the solutions of Kerr family. In the present paper we discuss the gravitational redshift for rotating body by using Kerr metric. The numerical calculations has been done under Newtonian approximation of angular momentum. It has been found that the value of gravitational redshift is influenced by the direction of spin of central body and also on the position (latitude) on the central body at which the photon is emitted. The variation of gravitational redshift from equatorial to non - equatorial region has been calculated and its implications are discussed in detail. I. INTRODUCTION from principle of equivalence. Snider in 1972 has mea- sured the redshift of the solar potassium absorption line General relativity is not only relativistic theory of grav- at 7699 A˚ by using an atomic - beam resonance - scat- itation proposed by Einstein, but it is the simplest theory tering technique [5]. Krisher et al. in 1993 had measured that is consistent with experimental data. Predictions of the gravitational redshift of Sun [6]. -
Eternal Inflation and Its Implications
IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 40 (2007) 6811–6826 doi:10.1088/1751-8113/40/25/S25 Eternal inflation and its implications Alan H Guth Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA E-mail: [email protected] Received 8 February 2006 Published 6 June 2007 Online at stacks.iop.org/JPhysA/40/6811 Abstract Isummarizetheargumentsthatstronglysuggestthatouruniverseisthe product of inflation. The mechanisms that lead to eternal inflation in both new and chaotic models are described. Although the infinity of pocket universes produced by eternal inflation are unobservable, it is argued that eternal inflation has real consequences in terms of the way that predictions are extracted from theoretical models. The ambiguities in defining probabilities in eternally inflating spacetimes are reviewed, with emphasis on the youngness paradox that results from a synchronous gauge regularization technique. Although inflation is generically eternal into the future, it is not eternal into the past: it can be proven under reasonable assumptions that the inflating region must be incomplete in past directions, so some physics other than inflation is needed to describe the past boundary of the inflating region. PACS numbers: 98.80.cQ, 98.80.Bp, 98.80.Es 1. Introduction: the successes of inflation Since the proposal of the inflationary model some 25 years ago [1–4], inflation has been remarkably successful in explaining many important qualitative and quantitative properties of the universe. In this paper, I will summarize the key successes, and then discuss a number of issues associated with the eternal nature of inflation. -
Abstract Generalized Natural Inflation and The
ABSTRACT Title of dissertation: GENERALIZED NATURAL INFLATION AND THE QUEST FOR COSMIC SYMMETRY BREAKING PATTERS Simon Riquelme Doctor of Philosophy, 2018 Dissertation directed by: Professor Zackaria Chacko Department of Physics We present a two-field model that generalizes Natural Inflation, in which the inflaton is the pseudo-Goldstone boson of an approximate symmetry that is spon- taneously broken, and the radial mode is dynamical. Within this model, which we designate as \Generalized Natural Inflation”, we analyze how the dynamics fundamentally depends on the mass of the radial mode and determine the size of the non-Gaussianities arising from such a scenario. We also motivate ongoing research within the coset construction formalism, that aims to clarify how the spontaneous symmetry breaking pattern of spacetime, gauge, and internal symmetries may allow us to get a deeper understanding, and an actual algebraic classification in the spirit of the so-called \zoology of con- densed matter", of different possible \cosmic states", some of which may be quite relevant for model-independent statements about different phases in the evolution of our universe. The outcome of these investigations will be reported elsewhere. GENERALIZED NATURAL INFLATION AND THE QUEST FOR COSMIC SYMMETRY BREAKING PATTERNS by Simon Riquelme Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2018 Advisory Committee: Professor Zackaria Chacko, Chair/Advisor Professor Richard Wentworth, Dean's Representative Professor Theodore Jacobson Professor Rabinda Mohapatra Professor Raman Sundrum A mis padres, Kattya y Luis. ii Acknowledgments It is a pleasure to acknowledge my adviser, Dr. -
Hubble's Evidence for the Big Bang
Hubble’s Evidence for the Big Bang | Instructor Guide Students will explore data from real galaxies to assemble evidence for the expansion of the Universe. Prerequisites ● Light spectra, including graphs of intensity vs. wavelength. ● Linear (y vs x) graphs and slope. ● Basic measurement statistics, like mean and standard deviation. Resources for Review ● Doppler Shift Overview ● Students will consider what the velocity vs. distance graph should look like for 3 different types of universes - a static universe, a universe with random motion, and an expanding universe. ● In an online interactive environment, students will collect evidence by: ○ using actual spectral data to calculate the recession velocities of the galaxies ○ using a “standard ruler” approach to estimate distances to the galaxies ● After they have collected the data, students will plot the galaxy velocities and distances to determine what type of model Universe is supported by their data. Grade Level: 9-12 Suggested Time One or two 50-minute class periods Multimedia Resources ● Hubble and the Big Bang WorldWide Telescope Interactive Materials ● Activity sheet - Hubble’s Evidence for the Big Bang Lesson Plan The following represents one manner in which the materials could be organized into a lesson: Focus Question: ● How does characterizing how galaxies move today tell us about the history of our Universe? Learning Objective: ● SWBAT collect and graph velocity and distance data for a set of galaxies, and argue that their data set provides evidence for the Big Bang theory of an expanding Universe. Activity Outline: 1. Engage a. Invite students to share their ideas about these questions: i. Where did the Universe come from? ii.