Cosmology for Everyone
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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 . -
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 = . -
Cosmic Background Explorer (COBE) and Beyond
From the Big Bang to the Nobel Prize: Cosmic Background Explorer (COBE) and Beyond Goddard Space Flight Center Lecture John Mather Nov. 21, 2006 Astronomical Search For Origins First Galaxies Big Bang Life Galaxies Evolve Planets Stars Looking Back in Time Measuring Distance This technique enables measurement of enormous distances Astronomer's Toolbox #2: Doppler Shift - Light Atoms emit light at discrete wavelengths that can be seen with a spectroscope This “line spectrum” identifies the atom and its velocity Galaxies attract each other, so the expansion should be slowing down -- Right?? To tell, we need to compare the velocity we measure on nearby galaxies to ones at very high redshift. In other words, we need to extend Hubble’s velocity vs distance plot to much greater distances. Nobel Prize Press Release The Royal Swedish Academy of Sciences has decided to award the Nobel Prize in Physics for 2006 jointly to John C. Mather, NASA Goddard Space Flight Center, Greenbelt, MD, USA, and George F. Smoot, University of California, Berkeley, CA, USA "for their discovery of the blackbody form and anisotropy of the cosmic microwave background radiation". The Power of Thought Georges Lemaitre & Albert Einstein George Gamow Robert Herman & Ralph Alpher Rashid Sunyaev Jim Peebles Power of Hardware - CMB Spectrum Paul Richards Mike Werner David Woody Frank Low Herb Gush Rai Weiss Brief COBE History • 1965, CMB announced - Penzias & Wilson; Dicke, Peebles, Roll, & Wilkinson • 1974, NASA AO for Explorers: ~ 150 proposals, including: – JPL anisotropy proposal (Gulkis, Janssen…) – Berkeley anisotropy proposal (Alvarez, Smoot…) – Goddard/MIT/Princeton COBE proposal (Hauser, Mather, Muehlner, Silverberg, Thaddeus, Weiss, Wilkinson) COBE History (2) • 1976, Mission Definition Science Team selected by HQ (Nancy Boggess, Program Scientist); PI’s chosen • ~ 1979, decision to build COBE in-house at GSFC • 1982, approval to construct for flight • 1986, Challenger explosion, start COBE redesign for Delta launch • 1989, Nov. -
The Birth of Stars and Planets
Unit 6: The Birth of Stars and Planets This material was developed by the Friends of the Dominion Astrophysical Observatory with the assistance of a Natural Science and Engineering Research Council PromoScience grant and the NRC. It is a part of a larger project to present grade-appropriate material that matches 2020 curriculum requirements to help students understand planets, with a focus on exoplanets. This material is aimed at BC Grade 6 students. French versions are available. Instructions for teachers ● For questions and to give feedback contact: Calvin Schmidt [email protected], ● All units build towards the Big Idea in the curriculum showing our solar system in the context of the Milky Way and the Universe, and provide background for understanding exoplanets. ● Look for Ideas for extending this section, Resources, and Review and discussion questions at the end of each topic in this Unit. These should give more background on each subject and spark further classroom ideas. We would be happy to help you expand on each topic and develop your own ideas for your students. Contact us at the [email protected]. Instructions for students ● If there are parts of this unit that you find confusing, please contact us at [email protected] for help. ● We recommend you do a few sections at a time. We have provided links to learn more about each topic. ● You don’t have to do the sections in order, but we recommend that. Do sections you find interesting first and come back and do more at another time. ● It is helpful to try the activities rather than just read them. -
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. -
Throughout the Universe, Galaxies Are Rushing Away from Us – and from Each Other – at Tremendously High Speeds
Our Universe Began with a Bang Throughout the Universe, galaxies are rushing away from us – and from each other – at tremendously high speeds. This fact tells us that the Universe is expanding over time. Edwin Hubble (after whom the Hubble Space Telescope was named) first measured the expansion in 1929. Observatories of the Carnegie Institution of Washington Edwin Hubble This posed a big question. If we could run the cosmic movie backward in time, would everything in the Universe be crammed together in a blazing fireball – the starting point of Edwin Hubble & Proceedings of The National Academy of Sciences Hubble’s famous diagram showing the the Big Bang? A lot of scientific distance versus velocity of the galaxies he debate and many new theories observed. The farther away the galaxies, the faster they are moving, showing that the followed Hubble’s discovery. Universe is expanding. Among those in the front lines of the debate were physicists Ralph Alpher and Robert Herman. In 1948 they predicted that an afterglow of this fireball should still exist, though at a much lower temperature than at the time of the Big Bang. Here’s why: As the Universe Fun Fact: expands, the waves of heat About radiation from the Big Bang are 1% of the stretched out, and cool from “snow” you see visible energy to infrared and on broadcast TV then to microwave wavelengths. is caused by the Microwaves are just short- cosmic microwave wavelength radio waves, the same background. form of energy used in microwave ovens. The prediction of an afterglow could be tested! Scientists began building instruments to detect this “cosmic microwave background”, or CMB. -
How Supernovae Became the Basis of Observational Cosmology
Journal of Astronomical History and Heritage, 19(2), 203–215 (2016). HOW SUPERNOVAE BECAME THE BASIS OF OBSERVATIONAL COSMOLOGY Maria Victorovna Pruzhinskaya Laboratoire de Physique Corpusculaire, Université Clermont Auvergne, Université Blaise Pascal, CNRS/IN2P3, Clermont-Ferrand, France; and Sternberg Astronomical Institute of Lomonosov Moscow State University, 119991, Moscow, Universitetsky prospect 13, Russia. Email: [email protected] and Sergey Mikhailovich Lisakov Laboratoire Lagrange, UMR7293, Université Nice Sophia-Antipolis, Observatoire de la Côte d’Azur, Boulevard de l'Observatoire, CS 34229, Nice, France. Email: [email protected] Abstract: This paper is dedicated to the discovery of one of the most important relationships in supernova cosmology—the relation between the peak luminosity of Type Ia supernovae and their luminosity decline rate after maximum light. The history of this relationship is quite long and interesting. The relationship was independently discovered by the American statistician and astronomer Bert Woodard Rust and the Soviet astronomer Yury Pavlovich Pskovskii in the 1970s. Using a limited sample of Type I supernovae they were able to show that the brighter the supernova is, the slower its luminosity declines after maximum. Only with the appearance of CCD cameras could Mark Phillips re-inspect this relationship on a new level of accuracy using a better sample of supernovae. His investigations confirmed the idea proposed earlier by Rust and Pskovskii. Keywords: supernovae, Pskovskii, Rust 1 INTRODUCTION However, from the moment that Albert Einstein (1879–1955; Whittaker, 1955) introduced into the In 1998–1999 astronomers discovered the accel- equations of the General Theory of Relativity a erating expansion of the Universe through the cosmological constant until the discovery of the observations of very far standard candles (for accelerating expansion of the Universe, nearly a review see Lipunov and Chernin, 2012). -
The Big-Bang Theory AST-101, Ast-117, AST-602
AST-101, Ast-117, AST-602 The Big-Bang theory Luis Anchordoqui Thursday, November 21, 19 1 17.1 The Expanding Universe! Last class.... Thursday, November 21, 19 2 Hubbles Law v = Ho × d Velocity of Hubbles Recession Distance Constant (Mpc) (Doppler Shift) (km/sec/Mpc) (km/sec) velocity Implies the Expansion of the Universe! distance Thursday, November 21, 19 3 The redshift of a Galaxy is: A. The rate at which a Galaxy is expanding in size B. How much reader the galaxy appears when observed at large distances C. the speed at which a galaxy is orbiting around the Milky Way D. the relative speed of the redder stars in the galaxy with respect to the blues stars E. The recessional velocity of a galaxy, expressed as a fraction of the speed of light Thursday, November 21, 19 4 The redshift of a Galaxy is: A. The rate at which a Galaxy is expanding in size B. How much reader the galaxy appears when observed at large distances C. the speed at which a galaxy is orbiting around the Milky Way D. the relative speed of the redder stars in the galaxy with respect to the blues stars E. The recessional velocity of a galaxy, expressed as a fraction of the speed of light Thursday, November 21, 19 5 To a first approximation, a rough maximum age of the Universe can be estimated using which of the following? A. the age of the oldest open clusters B. 1/H0 the Hubble time C. the age of the Sun D. -
A New Universe to Discover: a Guide to Careers in Astronomy
A New Universe to Discover A Guide to Careers in Astronomy Published by The American Astronomical Society What are Astronomy and Astrophysics? Ever since Galileo first turned his new-fangled one-inch “spyglass” on the moon in 1609, the popular image of the astronomer has been someone who peers through a telescope at the night sky. But astronomers virtually never put eye to lens these days. The main source of astronomical data is still photons (particles of light) from space, but the tools used to gather and analyze them are now so sophisticated that it’s no longer necessary (or even possible, in most cases) for a human eye to look through them. But for all the high-tech gadgetry, the 21st-Century astronomer is still trying to answer the same fundamental questions that puzzled Galileo: How does the universe work, and where did it come from? Webster’s dictionary defines “astronomy” as “the science that deals with the material universe beyond the earth’s atmosphere.” This definition is broad enough to include great theoretical physicists like Isaac Newton, Albert Einstein, and Stephen Hawking as well as astronomers like Copernicus, Johanes Kepler, Fred Hoyle, Edwin Hubble, Carl Sagan, Vera Rubin, and Margaret Burbidge. In fact, the words “astronomy” and “astrophysics” are pretty much interchangeable these days. Whatever you call them, astronomers seek the answers to many fascinating and fundamental questions. Among them: *Is there life beyond earth? *How did the sun and the planets form? *How old are the stars? *What exactly are dark matter and dark energy? *How did the Universe begin, and how will it end? Astronomy is a physical (non-biological) science, like physics and chemistry. -
The Big Bang
15- THE BIG ORIGINS BANGANAND NARAYANAN The Big Bang theory “Who really knows, who can declare? The scientific study of the origin and currently occupies When it started or where from? evolution of the universe is called the center stage in From when and where this creation has arisen cosmology. From the early years of the modern cosmology. 20th century, the observational aspects Perhaps it formed itself, perhaps it did not.” And yet, half a of scientific cosmology began to gather century ago, it was – Rig Veda (10:129), 9th century BCE. wide attention. This was because of only one among a a small series of startling discoveries few divergent world made by scientists like Edwin Hubble views, with competing hese words were written over 2000 years that changed the way we understood claims to the truth. ago. And yet, if they sound contemporary, it the physical universe. This article highlights Tis because they echo a trait that is distinctly a few trail-blazing human; the highly evolved ability of our species to Hubble had an advantage which few events in the journey look at the world around us and wonder how it all astronomers in the early 20th century of the Big Bang model came to be. How often have you asked these very had – access to the Mount Wilson from the fringes to same questions? Observatory in California. This the centerfold of particular observatory housed the astrophysics, a We assume that all things have a beginning. Could this be true for this vastly complex universe as well? largest telescopes of that time and journey that is yielded high quality data. -
Edwin Hubble (1889-1953) Measured Distance to Andromeda Galaxy (M31) • Noticed Individual Bright Stars in Andromeda
The Resolution: Edwin Hubble (1889-1953) Measured Distance to Andromeda Galaxy (M31) • Noticed individual bright stars in Andromeda • Calculated the distance using the Period-Luminosity Relation for Cepheid Variable Stars The Resolution: Edwin Hubble (1889-1953) Measured Distance to Andromeda Galaxy (M31) • Noticed individual bright stars in Andromeda • Calculated the distance using the Period-Luminosity Relation for Cepheid Variable Stars 1. MEASURED magnitude & Period The Resolution: Edwin Hubble (1889-1953) Measured Distance to Andromeda Galaxy (M31) • Noticed individual bright stars in Andromeda • Calculated the distance using the Period-Luminosity Relation for Cepheid Variable Stars 1. MEASURED magnitude & Period 2. CALCULATED Luminosity The Resolution: Edwin Hubble (1889-1953) Measured Distance to Andromeda Galaxy (M31) • Noticed individual bright stars in Andromeda • Calculated the distance using the Period-Luminosity Relation for Cepheid Variable Stars 1. MEASURED magnitude & Period 2. CALCULATED Luminosity 3. CALCULATED Distance The Resolution: Edwin Hubble (1889-1953) Measured Distance to Andromeda Galaxy (M31) • Noticed individual bright stars in Andromeda • Calculated the distance using the Period-Luminosity Relation for Cepheid Variable Stars 1. MEASURED magnitude & Period 2. CALCULATED Luminosity 3. CALCULATED Distance • Result was MUCH farther than expected... • (2.3 Million Ly - well beyond Milky Way) Henrietta Leavitt & Period-Luminosity Relation Cepheid Variables: Bright stars whose Luminosity (energy output) varies every -
Finding the Radiation from the Big Bang
Finding The Radiation from the Big Bang P. J. E. Peebles and R. B. Partridge January 9, 2007 4. Preface 6. Chapter 1. Introduction 13. Chapter 2. A guide to cosmology 14. The expanding universe 19. The thermal cosmic microwave background radiation 21. What is the universe made of? 26. Chapter 3. Origins of the Cosmology of 1960 27. Nucleosynthesis in a hot big bang 32. Nucleosynthesis in alternative cosmologies 36. Thermal radiation from a bouncing universe 37. Detecting the cosmic microwave background radiation 44. Cosmology in 1960 52. Chapter 4. Cosmology in the 1960s 53. David Hogg: Early Low-Noise and Related Studies at Bell Lab- oratories, Holmdel, N.J. 57. Nick Woolf: Conversations with Dicke 59. George Field: Cyanogen and the CMBR 62. Pat Thaddeus 63. Don Osterbrock: The Helium Content of the Universe 70. Igor Novikov: Cosmology in the Soviet Union in the 1960s 78. Andrei Doroshkevich: Cosmology in the Sixties 1 80. Rashid Sunyaev 81. Arno Penzias: Encountering Cosmology 95. Bob Wilson: Two Astronomical Discoveries 114. Bernard F. Burke: Radio astronomy from first contacts to the CMBR 122. Kenneth C. Turner: Spreading the Word — or How the News Went From Princeton to Holmdel 123. Jim Peebles: How I Learned Physical Cosmology 136. David T. Wilkinson: Measuring the Cosmic Microwave Back- ground Radiation 144. Peter Roll: Recollections of the Second Measurement of the CMBR at Princeton University in 1965 153. Bob Wagoner: An Initial Impact of the CMBR on Nucleosyn- thesis in Big and Little Bangs 157. Martin Rees: Advances in Cosmology and Relativistic Astro- physics 163.