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Introductory Particle Cosmology Introductory Particle Cosmology Luca Doria Institut für Kernphysik Johannes-Gutenberg Universität Mainz Lecture 1 Introduction Mathematical Tools Generic Vector Bases Observation Basis Change Cosmology CMB Vectors in Curved Geometries Cosmological Principle Structure Formation Tensors, Metric Tensor FLRW Metric Red-shift/Distance (Riemann) Manifolds Friedmann Equations Connection Cosmic Distances Geodesics and Curvature Cosmological Models Particle Cosmology General Relativity Dark Matter (Models + Exp.) Equivalence Principle Dark Energy (Models + Exp) Einstein Equations Inflationary Models Standard Model Gravitational Waves Density Perturbations Brief History Particle Content Gauge Principle, CPV, Strong CP EW Symmetry Breaking Beyond the SM Notes and Slides: www.staff.uni-mainz.de/doria/partcosm.html Sommersemester 2018 Luca Doria, JGU Mainz 2 Plan Datum Von Bis Raum 1 Di, 17. Apr. 2018 10:00 12:00 05 119 Minkowski-Raum 2 Do, 19. Apr. 2018 08:00 10:00 05 119 Minkowski-Raum 3 Di, 24. Apr. 2018 10:00 12:00 05 119 Minkowski-Raum 4 Do, 26. Apr. 2018 08:00 10:00 05 119 Minkowski-Raum 5 Do, 3. Mai 2018 08:00 10:00 05 119 Minkowski-Raum 6 Di, 8. Mai 2018 10:00 12:00 05 119 Minkowski-Raum 7 Di, 15. Mai 2018 10:00 12:00 05 119 Minkowski-Raum 8 Do, 17. Mai 2018 08:00 10:00 05 119 Minkowski-Raum H.Minkowski 9 Di, 22. Mai 2018 10:00 12:00 05 119 Minkowski-Raum (1864-1909) 10 Do, 24. Mai 2018 08:00 10:00 05 119 Minkowski-Raum 11 Di, 29. Mai 2018 10:00 12:00 05 119 Minkowski-Raum 12 Di, 5. Jun. 2018 10:00 12:00 05 119 Minkowski-Raum 13 Do, 7. Jun. 2018 08:00 10:00 05 119 Minkowski-Raum 14 Di, 12. Jun. 2018 10:00 12:00 05 119 Minkowski-Raum 15 Do, 14. Jun. 2018 08:00 10:00 05 119 Minkowski-Raum 16 Di, 19. Jun. 2018 10:00 12:00 05 119 Minkowski-Raum 17 Do, 21. Jun. 2018 08:00 10:00 05 119 Minkowski-Raum 18 Di, 26. Jun. 2018 10:00 12:00 05 119 Minkowski-Raum 19 Do, 28. Jun. 2018 08:00 10:00 05 119 Minkowski-Raum 20 Di, 3. Jul. 2018 10:00 12:00 05 119 Minkowski-Raum 21 Do, 5. Jul. 2018 08:00 10:00 05 119 Minkowski-Raum Physics Dept. Building, 5th Floor Sommersemester 2018 Luca Doria, JGU Mainz Cosmic History Sommersemester 2018 Luca Doria, JGU Mainz 4 (Ancient) History <16th century BC, Mesopotamian: Flat, circular Earth in the middle of an ocean 4th century BC, Aristotle: Earth-centric, finite, immutable universe 3th century BC, Archimedes “measured” the diameter of the Universe (2lyr !). 2th century BC, Ptolemy: Earth-Centred universe, Sun and planets revolving Non-European astronomers proposing Sun-centered theories Sommersemester 2018 Luca Doria, JGU Mainz 5 (non-BC) History 1540s: Copernicus proposes the heliocentric theory 1584: Giordano Bruno proposes a non-privileged position of the Sun in the cosmos. 1600s: Kepler discovered his laws and elliptic motion. He believed in a finite universe 1680s: Isaac Newton: Theory if Gravitation Sommersemester 2018 Luca Doria, JGU Mainz 6 (non-BC) History 1755: I. Kant argues that nebulae are really separate galaxies 1826: Olber’s Paradox 1837: Bessel successfully measures the first parallax 1915: A. Einstein publishes the General Relativity Theory Sommersemester 2018 Luca Doria, JGU Mainz 7 (after GR) History 1912: Henrietta Leavitt discovers the Cepheid variable stars 1922: Friedmann finds expanding solutions of GR 1923: E. Hubble measures an apparent expansion of the universe 1933: E. Milne states the Cosmological Principle 1948: G. Gamow predicts the CMB Sommersemester 2018 Luca Doria, JGU Mainz 8 (after GR) History 1950: F. Hoyle coins the word “Big-Bang” 1965: A. Penzias and R. Wilson discover the CMB 1967: A. Sakharov states the requirements for baryogengesis 1970: V. Rubin and K. Ford present precise data on galaxy rotational curves Sommersemester 2018 Luca Doria, JGU Mainz 9 (Modern) History 1980: A. Guth presents the idea of cosmic inflation 1982: J.Peebles and others propose CDM 1990s: COBE Mission and the first measurement of CMB anisotropy 1996: Hubble Space Telescope deep-field pictures 1998: Supernova Cosmology Project and High-Z Supernova Search Sommersemester 2018 Luca Doria, JGU Mainz 10 (Modern) History 1999: More COBE data and BOOMERanG experiment 2003-2011: WMAP, Planck and LambdaCDM 2014: BICEP2 and (the not confirmed) B-modes 2016: First Measurement on Earth of Gravitational Waves Sommersemester 2018 Luca Doria, JGU Mainz 11 Successes and Issues Successes of the Standard Model of Cosmology - Explains the dark sky (Olber’s Paradox) - Expansion of the Universe - Approximately correct age of the Universe vs oldest stars - Existence of the Cosmic Microwave Background (CMB) - Primordial nuclear abundances Issues: - Baryon asymmetry - Flatness - Homogeneity - Origin of the primordial fluctuations - … Sommersemester 2018 Luca Doria, JGU Mainz 12 Successes and Issues Successes of the Standard Model of Cosmology - Explains the dark sky (Olber’s Paradox) - Expansion of the Universe - Approximately correct age of the Universe vs oldest stars - Existence of the Cosmic Microwave Background (CMB) - Primordial nuclear abundances Issues: - Baryon asymmetry - Flatness - Homogeneity - Origin of the primordial fluctuations - … Inflation as possible solution Sommersemester 2018 Luca Doria, JGU Mainz 12 Particle Cosmology Planck Scale LHC Collisions Nuclei Atoms Our Size Galaxy Clusters Size of the Universe Sommersemester 2018 Luca Doria, JGU Mainz 13 Particle Cosmology Planck Scale LHC Collisions Nuclei Atoms Our Size Galaxy Clusters Size of the Universe Sommersemester 2018 Luca Doria, JGU Mainz 13 Part 1: Mathematical Tools Vector Spaces (Ortho-normal) bases Change of Basis Non-orthonormal bases Covariant and contravariant vectors Tensors Curved Spaces Vectors on curved spaces Covariant Differentiation Geodetics Curvature Sommersemester 2018 Luca Doria, JGU Mainz 14 NATURAL SYSTEM OF UNITS IN GENERAL RELATIVITY ALAN L. MYERS Abstract. The international system (SI) of units based on the kilogram, meter, and second is inconvenient for computations involving the masses of stars or the size of a galaxy. Papers about quantum gravity express mass, length, and time in terms of energy, usually powers of GeV. Geometrized units, in which all units are expressed in terms of powers of length are also prevalent in the literature of general relativity. Here equations are provided for conversions from geometrized or natural units to SI units in order to make numerical calculations. 1. International System of Units The International System of Units (SI) is based on the meter-kilogram-second (MKS) system of units. A first course in any field of science or engineering usually begins with a discussion of SI units. Given an equation, students learn to check instinctively the units of the various terms for consistency. For example, in SI units for Newton’s law of universal gravitation GmM F = r2 m and M are the masses of the two bodies in kilograms, r is the distance between their centers in 2 meters, F is the force of attraction in newtons (1 N = 1 kg m s− ), and G = Newton’s constant 11 3 1 2 =6.6743 10− m kg− s− . This equation satisfies the consistency test because the terms on ⇥ 2 both sides of the equation have the same units (kg m s− ). Cosmologists toss the SI system of units out the window so that every variable is expressed in powers of energy. The equations are simplified by the absence of constants including Newton’s constant (G) and the speed of light (c). However, for a person educated in the SI system of units, the lingo of “natural units” used by cosmologists is confusing and seems (incorrectly) to display a casual disregard of the importance of units in calculations. First let us examine units term-by-term in the Einstein equation of general relativity 1 8⇡G (1) R Rg + ⇤g = T µ⌫ − 2 µ⌫ µ⌫ c4 µ⌫ written in SI units. Popular articles and most textbooks on general relativity introduce this equa- tion without discussing its units. The energy-momentum tensor Tµ⌫ has units of energy density 3 1 2 (J m− ) or, equivalently, momentum flux density (kg m− s− ). Multiplication of the units of 1 2 4 2 1 1 2 Tµ⌫ (kg m− s− ) by the units of G/c (s kg− m− ) yields units of m− for the RHS of the 2 equation. The terms on the LHS must have the same units (m− ). The metric tensor (gµ⌫ )is dimensionless. The Riemann tensor is the second derivative with respect to distance of the metric 2 tensor and therefore has units of m− . The index-lowered forms of the Riemann tensor, Rµ⌫ and R, 2 have units of m− so the units are the same on both sides of the equation, as required. 2. Natural units. So-called natural units are used almost exclusively in cosmology and general relativity. In order to read the literature, it is necessary to learn2ALANL.MYERS how to write equations…but and perform first calculations some in Physics natural units. In the version of natural units used in cosmology, four fundamental constants are set to unity: where 8 Natural Units c = ~ = ✏ = kB =1 c = speed of light = 2.9979 10 m/s ◦ ⇥ 1 34 ~ = reduced Planck constant = 1.0546 10− Js ⇥ 12 2 4 1 3 ✏ = electric constant = 8.8542 10− A s kg− m− ◦ ⇥ 23 1 k = Boltzmann constant = 1.3806 10− JK− B ⇥ 8 1 34 As a consequence of these definitions, 1 s = 2.9979 10 m and 1 s− =1.0546 10− J. Length and time acquire the units of reciprocal energy; energy⇥ and mass have the same units.⇥ Any kinematical Consequences of thisvariable choice: with 1 SIs =2 units.9979 of (kg↵ 10mβ8smγ )maybeexpressedinSIunits: ⇥ 1 34 ↵ β γ β+γ β 2↵ (2) 1s− =1.0546− J (E) − − ~ c − 10 where E in an arbitrarily chosen energy unit.
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