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 Connection Cosmic Distances Geodesics and Curvature Cosmological Models

Particle Cosmology

General Relativity Dark Matter (Models + Exp.) Equivalence Principle Dark (Models + Exp) Einstein Equations Inflationary Models Standard Model Gravitational Waves 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, 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

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 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 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 is inconvenient for computations involving the of stars or the size of a galaxy. Papers about quantum express , , and 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 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 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 (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- tensor Tµ⌫ has units of 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 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↵ 10m8sm )maybeexpressedinSIunits: ⇥ 1 34 ↵ + 2↵ (2) 1s =1.0546 J (E) ~ c

10 where E in an arbitrarily chosen energy unit. A popular choice is GeV (1 GeV = 1.6022 10 J). ↵ ⇥ Setting the constants ~ and c equal to unity gives natural units of GeV . Given natural units Let’s try to measure everything↵ with Energy! of GeV , the desired SI unit can always be recovered by multiplying by the conversion factor + 2↵ 1 This means: [time]=[(~lengthc ]=[).Energy] 1 Consider, for example, momentum with SI units of (kg m s ). For ↵ = 1, = 1, and = 1, ↵ 1 1 the natural unit is GeV =Gev and the conversion factor is c . Conversion from natural to SI units gives: What about , momentum, mass,…? 10 E 1.6022 10 J 19 1 1 GeV = = ⇥ =5.3444 10 kg m s c 2.9979 108 ms 1 ⇥ ⇥ Table 1 provides conversion factors for some of the variables encountered in cosmology. Energy, Sommersemester 2018 mass, and momentumLuca Doria, have JGU the Mainz same natural units. Velocity, ,15 and charge are dimensionless. Note that (force per unit area) has the same units as energy density in both systems of units, as it must. In the system of natural units, the factors in Table 1 are unity by definition. To make conversions, cosmologists multiply or divide by the factors ~ and c with appropriate exponents to obtain the units desired. This seems like a trial-and-error procedure! Rules for converting units for SI to natural units, and for the reverse conversion, are given in Table 1.

+ 2↵ Table 1. Natural Units. Factor is (~ c ). To convert natural unit SI, multiply by factor. To convert SI natural unit, divide by factor. For SI units! of ↵ ↵ ! (kg m s ), natural units are E .

Variable SI Unit Natural Unit Factor Natural unit SI unit 2 ! 27 mass kg E c 1 GeV 1.7827 10 kg 1 1 ! ⇥ 16 length m E ~c 1 GeV 1.9733 10 m 1 1 ! ⇥ 25 time s E ~ 1 GeV 6.5823 10 s 2 2 ! ⇥ 10 energy kg m s E11GeV1.6022 10 J 1 1 ! ⇥ 19 1 momentum kg m s E c 1 GeV 5.3444 10 kg m s 1 ! ⇥ 8 1 velocity m s dimensionless c 1 2.9979 10 ms 2 1 ! ⇥ 34 angular momentum kg m s dimensionless ~ 1 1.0546 10 Js 2 2 2 2 ! ⇥ 32 2 area m E (~c) 1 GeV 3.8938 10 m 2 2 1 2 ! ⇥ 5 force kg m s E (~c) 1 GeV 8.1194 10 N 1 2 4 3 4 ! ⇥ 37 3 energy density kg m s E (~c) 1 GeV 2.0852 10 Jm ! ⇥ 19 charge C = A s dimensionless 1 1 5.2909 10 C · ! ⇥

The entry in Table 1 for charge requires an explanation. The dimensionless fine-structure constant (↵)is: e2 ↵ = =0.0072974 4⇡✏ ~c Conversions

Conversion form SI SI units are combinations of mass/length/time units. We would like to convert everything in energy, eventually setting hbar=c=1

↵ a b c kg m s = E ~ c

Converting E, hbar, c in SI units and comparing the exponents, we obtain

↵ ↵ + 2↵ kg m s = E ~ c

Common choice for [E] is GeV (1.6022x10-10 J)

Sommersemester 2018 Luca Doria, JGU Mainz 16 2ALANL.MYERS

where c = speed of light = 2.9979 108 m/s ⇥ 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 variable with SI units of (kg↵ m s )maybeexpressedinSIunits:

↵ + 2↵ (2) (E) ~ c

10 where E in an arbitrarily chosen energy unit. A popular choice is GeV (1 GeV = 1.6022 10 J). ↵ ⇥ Setting the constants ~ and c equal to unity gives natural units of GeV . Given natural units ↵ of GeV , the desired SI unit can always be recovered by multiplying by the conversion factor + 2↵ (~ c ). 1 Consider, for example, momentum with SI units of (kg m s ). For ↵ = 1, = 1, and = 1, ↵ 1 1 the natural unit is GeV =Gev and the conversion factor is c . Conversion from natural to SI units gives: 10 E 1.6022 10 J 19 1 1 GeV = = ⇥ =5.3444 10 kg m s c 2.9979 108 ms 1 ⇥ ⇥ Table 1 provides conversion factors for some of the variables encountered in cosmology. Energy, mass, and momentum have the same natural units. Velocity, angular momentum, and charge are dimensionless. Note that pressure (force per unit area) has the same units as energy density in both systems of units, as it must. In the system of natural units, the factors in Table 1 are unity by definition. To make conversions, cosmologists multiply or divide by the factors ~ and c with appropriate exponents to obtain the units desired. This seems like a trial-and-error procedure! Rules for converting units for SI to natural units, and for the reverse conversion, are given in Table 1.

+ 2↵ Table 1. Natural Units. Factor is (~ c ). To convert naturalConversions unit SI, multiply by factor. To convert SI natural unit, divide by factor. For SI units! of ↵ ↵ ! (kg m s ), natural units are E .

Variable SI Unit Natural Unit Factor Natural unit SI unit 2 ! 27 mass kg E c 1 GeV 1.7827 10 kg 1 1 ! ⇥ 16 length m E ~c 1 GeV 1.9733 10 m 1 1 ! ⇥ 25 time s E ~ 1 GeV 6.5823 10 s 2 2 ! ⇥ 10 energy kg m s E11GeV1.6022 10 J 1 1 ! ⇥ 19 1 momentum kg m s E c 1 GeV 5.3444 10 kg m s 1 ! ⇥ 8 1 velocity m s dimensionless c 1 2.9979 10 ms 2 1 ! ⇥ 34 angular momentum kg m s dimensionless ~ 1 1.0546 10 Js 2 2 2 2 ! ⇥ 32 2 area m E (~c) 1 GeV 3.8938 10 m 2 2 1 2 ! ⇥ 5 force kg m s E (~c) 1 GeV 8.1194 10 N 1 2 4 3 4 ! ⇥ 37 3 energy density kg m s E (~c) 1 GeV 2.0852 10 Jm ! ⇥ 19 charge C = A s dimensionless 1 1 5.2909 10 C · ! ⇥

The entry in Table 1 for charge requires an explanation. The dimensionless fine-structure constant (↵)is: e2 ↵ = =0.0072974 4⇡✏ ~c

↵ 2↵ Note: Geometrized unit system: G=c=1 (everything in length units, conv.factor= G c ) (G=c=habar=1: everything in units of Planck scales, see next slide)

Sommersemester 2018 Luca Doria, JGU Mainz 17 Planck Scale

The fundamental constants of Nature set a natural scale for the measurement units:

~G 35 Planck length: lP = 3 1.6 10 m r c ⇠ ⇥

~c 8 Planck Mass: mP = 2.1 10 kg r G ⇠ ⇥

~G 44 Planck Time: t = 5.4 10 s P 5 ⇠ ⇥ r c

5 ~c 9 19 Planck Energy: EP = 1.96 10 J 1.22 10 GeV r G ⇠ ⇥ ⇠ ⇥

..etc…

Sommersemester 2018 Luca Doria, JGU Mainz 18