Particles in the Dark Universe: A Student’s Guide to Particle Physics and Cosmology

Foreword to the First Edition
Preface to the Second Edition
Preface to the First Edition
	References
Acknowledgments
Contents
About the Author
1 Introduction
	1.1 The First Dark Matter Paper
		1.1.1 Prehistory
		1.1.2 The Galactic Scale
		1.1.3 Stabilization of the Structures
	1.2 Local Dark Matter
	1.3 Anomalies in Rotation Curves of Galaxies
	1.4 Cluster Dark Matter
	1.5 Gravitational Lensing
	1.6 Bullet Cluster
	1.7 Compared of Three Matter Abundance
	1.8 Cosmic Microwave Background (CMB)
	1.9 Alternatives
	References
Part I The Primordial Universe
	2 Inflation and Reheating [MP →TRH]
		2.1 The Context
			2.1.1 The Hubble Law
			2.1.2 The Friedmann Equations in a Dust Universe
				2.1.2.1 The Hubble Parameter
				2.1.2.2 The Continuity Equation
			2.1.3 The Friedmann Equations in a Radiative Universe
			2.1.4 The Friedmann–Lemaitre–Robertson–Walker (FLRW) Metric
				2.1.4.1 Generalities
				2.1.4.2 Geometry of the Universe
				2.1.4.3 Redshift
				2.1.4.4 Hubble Law
				2.1.4.5 Measuring the Size of the Universe
			2.1.5 Friedmann Equation in General Relativity
				2.1.5.1 The Friedmann Equations
				2.1.5.2 The Deceleration Equation
				2.1.5.3 The Cosmological Constant Case
			2.1.6 Another Look on the Hubble Expansion
			2.1.7 Historical Perspectives on Cosmological Universes*
				2.1.7.1 The Oblers\' Paradox
				2.1.7.2 The Mach\'s Principle
				2.1.7.3 The Einstein Universe (1917)
				2.1.7.4 The de Sitter Universe (1917)
				2.1.7.5 The Friedmann Universe (1922)
				2.1.7.6 The Friedmann–Einstein Universe (1931)
				2.1.7.7 The Einstein–de Sitter Universe (1932)
				2.1.7.8 The Milne Universe (1932)
				2.1.7.9 The Tolman (Cyclic) Universe (1931)
				2.1.7.10 The Steady-State Universe
				2.1.7.11 Summary: Teaching Universes
			2.1.8 The Comoving Distance or Codistance
				2.1.8.1 Generalities
				2.1.8.2 The Size of the Universe (Bis)
		2.2 Inflation [10-43-10-37Seconds]
			2.2.1 The Horizon Problem
			2.2.2 The Flatness Problem
			2.2.3 The Inflaton
			2.2.4 The Equation of Motion
			2.2.5 The Equation of Motion (Generalization)
			2.2.6 The Slow-Roll Regime
				2.2.6.1 The Context
				2.2.6.2 The V = 12m2ϕ2 Case
			2.2.7 Some Models*
				2.2.7.1 Starobinsky Model
				2.2.7.2 T-Attractor Models
				2.2.7.3 Constraints
			2.2.8 The Coherent Oscillation Regime
			2.2.9 The General Case, V(ϕ)
			2.2.10 Constraint from Perturbations*
				2.2.10.1 Generalities
				2.2.10.2 In an Expanding Universe
			2.2.11 Preheating and Dark Matter*
				2.2.11.1 Parametric Resonance
				2.2.11.2 Narrow Resonance Interpreted as Bose Condensates
				2.2.11.3 Production of Dark Matter in the Preheating Era
			2.2.12 Inflation: Summary and Tools for an IntroductoryLecture
				2.2.12.1 Solving the Horizon Problem
				2.2.12.2 Amplitude of the Perturbations, As
				2.2.12.3 Tilt of the Spectrum, ns
		2.3 Reheating: Non-thermal Phase [10-37 - 10-30Seconds]
			2.3.1 The Context
				2.3.1.1 Boltzmann Equation for the Dust (Inflaton or Non-relativistic Fields)
				2.3.1.2 Boltzmann Equation for the Radiation (Relativistic Fields)
				2.3.1.3 The Influence of the Nature of the Inflaton
			2.3.2 The (Non-thermal) Distribution Function
				2.3.2.1 Time Evolution of the Densities
				2.3.2.2 The Matter
				2.3.2.3 The Radiation
				2.3.2.4 The Scale Factor
				2.3.2.5 Summary
				2.3.2.6 The Distribution Function
			2.3.3 End of the Thermalization Process: Transition Toward a Thermal Bath
				2.3.3.1 Understanding the Process
				2.3.3.2 Computing the End of Thermalization Process
			2.3.4 Dark Matter Production During the Non-thermal Phase of the Reheating
				2.3.4.1 The Context
				2.3.4.2 Direct Production by Inflaton Decay
				2.3.4.3 Production by Scattering
			2.3.5 Production of (Dark) Matter from Inflaton Scattering
				2.3.5.1 Production from a Direct Coupling: L=σϕ2 X2
				2.3.5.2 Production of (Dark) Matter from Gravitational Scattering*
				2.3.5.3 The Bogoliubov Approach*
				2.3.5.4 The Bogoliubov Approach in the Literature*
				2.3.5.5 Non-minimal Coupling to the Ricci Scalar R*
		2.4 Gravitational Production During Inflation*
		2.5 Reheating: Thermal Phase [10-30-10-28Seconds]
			2.5.1 Understanding the Reheating
			2.5.2 Non-instantaneous Reheating
				2.5.2.1 Evolution of the Temperature During Reheating
				2.5.2.2 A Closer Look on the Hubble Constant*
			2.5.3 Producing Dark Matter During the Reheating Phase
				2.5.3.1 The Context
				2.5.3.2 Production from Inflaton Decay
				2.5.3.3 Production by Scattering
		2.6 The Thermal Era [10-28-mχ]
			2.6.1 Instantaneous Reheating and Instantaneous Thermalization
				2.6.1.1 Radiation Dominated Universe
				2.6.1.2 Matter-Dominated Universe
		References
	3 A Thermal Universe [TRH →TCMB]
		3.1 Thermodynamics
			3.1.1 A Brief Thermal History of the Universe in Some Dates and Numbers
			3.1.2 Statistics of Gas, Pressure, and Radiation: The Classic Case
			3.1.3 Statistics of Gas, Pressure, and Radiation: The Quantum Case
				3.1.3.1 Distribution Functions and Thermodynamics Quantities
			3.1.4 In the Primordial Plasma
			3.1.5 Degrees of Freedom
				3.1.5.1 Computation of gρ(T)
				3.1.5.2 QCD (Quark-Hadron) Phase Transition
				3.1.5.3 A Little History of gρ(T): Summary
			3.1.6 Time and Temperature
			3.1.7 The Entropy
			3.1.8 The Meaning of Decoupling
		3.2 Chemical Decoupling or Kinetic/Thermal Decoupling?
			3.2.1 The Main Idea
			3.2.2 Approximate Solution*
			3.2.3 What Is Happening After the Decoupling?
			3.2.4 Transfer of Energy and Thermalization
				3.2.4.1 Generalities
				3.2.4.2 A Specific Case: Exchanged of a Massless Gauge Boson *
				3.2.4.3 Thermalization
				3.2.4.4 γ\' Entering in the Dance
		3.3 The Case of Light Species
			3.3.1 The Neutrino Decoupling
			3.3.2 The Tremaine-Gunn Bound
			3.3.3 Dark Radiation
				3.3.3.1 Generalities
				3.3.3.2 One Exemple to Increase Neff *
				3.3.3.3 Another Example: The Case of the Mirror Dark Matter *
			3.3.4 The Recombination: Decoupling of the Photons
				3.3.4.1 The Recombination
				3.3.4.2 The Last Scattering Surface
			3.3.5 The Dark Ages, or Re-ionization
		3.4 The Big-Bang Nucleosynthesis
			3.4.1 The Context
			3.4.2 Overview
			3.4.3 The Deuterium Formation
			3.4.4 The Lithium Problem
		3.5 Baryogenesis
			3.5.1 Baryon Asymmetry in the Universe
			3.5.2 Ingredients for a Successful Baryogenesis
				3.5.2.1 B Violation
				3.5.2.2 C and CP Violation
				3.5.2.3 Out-of-Equilibrium Condition
			3.5.3 Examples
				3.5.3.1 The Case of Decaying Particle
				3.5.3.2 Leptogenesis
			3.5.4 Sphalerons
		3.6 Producing Dark Matter in Thermal Equilibrium
			3.6.1 The Boltzmann Equation
			3.6.2 Overview
			3.6.3 Solving the Equation
				3.6.3.1 s-Wave
				3.6.3.2 General Solution
				3.6.3.3 Hot Dark Matter
				3.6.3.4 Another Approach to Average the Annihilation Cross Section
			3.6.4 The Lee-Weinberg Bound
			3.6.5 Two Exceptions to the Boltzmann Equation*
				3.6.5.1 The Pole Region
				3.6.5.2 Kinematic Threshold
		3.7 Non-thermal Production of Dark Matter
			3.7.1 The Idea
				3.7.1.1 Case a: Heavy Mediator, MH TRH
				3.7.1.2 Case b: Light Mediator, MH s, Weak-Like Coupling: α\' αEW
				3.7.1.3 Case c: Light Mediator, MH s, Feebly Like Coupling: α\' αEW
			3.7.2 Axion as a Dark Matter Candidate*
				3.7.2.1 The Thermal Production
				3.7.2.2 The Misalignment Mechanism
				3.7.2.3 QCD-Axion Dark Matter
			3.7.3 The Special Case of the Gravitino**
				3.7.3.1 What Is a Gravitino
				3.7.3.2 MSUSY < TRH
			3.7.4 Non-thermal Production Through Decays
				3.7.4.1 Generalities
				3.7.4.2 An Example: Decay of the Gravitino to Populate Dark Matter **
			3.7.5 Primordial Black Holes as Dark Matter
				3.7.5.1 Absorption
				3.7.5.2 Primordial Black Hole Formation and Distribution
				3.7.5.3 Primordial Black Hole as a Dark Matter Candidate
				3.7.5.4 The Unruh Effect
				3.7.5.5 Hawking Radiation
				3.7.5.6 Constraints
				3.7.5.7 Summary
				3.7.5.8 Can the PBHs Reheat the Universe?
				3.7.5.9 Can the PBHs Populate the Dark Matter?
				3.7.5.10 Stellar Black Holes
		3.8 Extracting Informations from the CMB Spectrum
			3.8.1 Generalities
			3.8.2 To Find the Components of the Universe
				3.8.2.1 Influence of the Matter, Ωm
				3.8.2.2 Influence of the Curvature, Ωk
				3.8.2.3 Influence of the Cosmological Constant, Ω
				3.8.2.4 Influence of the Baryons, Ωb
			3.8.3 H0 Tension and Early Dark Energy*
		References
Part II Modern Times [TCMB →T0]
	4 Direct Detection [T0]
		4.1 Generality
		4.2 Velocity Distribution of Dark Matter: f(v)
		4.3 Measuring a Differential Rate: d σd |q|2
			4.3.1 Kinematics
			4.3.2 Differential Rate
		4.4 Structure Function of the Nucleus: F(q)
		4.5 Computing a Rate
		4.6 Being More Realistic
			4.6.1 Taking into Account the Earth Velocity
			4.6.2 Annual Modulation of the Signal
		4.7 Influence of the Structure of the Nucleons**
		4.8 Spinorial Effect
		4.9 More About the Effective Approach
			4.9.1 Validity of the Approach
			4.9.2 Effective Operators
				4.9.2.1 Generalities
				4.9.2.2 Scalar Coefficient: Generalities
				4.9.2.3 Scalar Coefficient: Application
				4.9.2.4 Vector Coefficient: Generalities
				4.9.2.5 Vector Coefficient: Application
				4.9.2.6 Majorana Case
			4.9.3 Gluons and Heavy Quark Contributions*
		References
	5 In the Galaxies [T0]
		5.1 The Anatomy of the Milky Way
			5.1.1 Internal Characteristics
			5.1.2 The Color of the Sky: The Diffuse Gamma Ray Background
				5.1.2.1 X-Ray Diffuse Background
				5.1.2.2 Gamma-Ray Diffuse Background
			5.1.3 Galactic Coordinates, Velocity of the Sun and of the Earth
		5.2 Computation of a Flux
		5.3 Example of the Isothermal Profile
		5.4 Radiative Processes in Astrophysics Part I: The Nonrelativistic Case
			5.4.1 Maxwell Equations
			5.4.2 Loss of Energy of a Moving Charged Particle
				5.4.2.1 Larmor\'s Formula
				5.4.2.2 Case of a Rotating Particle
			5.4.3 Coulomb and Ionization Losses
			5.4.4 Thomson Scattering
			5.4.5 Cyclotron Radiation
			5.4.6 Bremsstrahlung Radiation
		5.5 Notions of Relativity
			5.5.1 Main Idea
			5.5.2 Lorentz Transformations
			5.5.3 Relativistic Larmor\'s Formula
			5.5.4 Doppler Effect
			5.5.5 Transformations on the Energies
			5.5.6 The Fizeau Experiment
			5.5.7 The Hafele–Keating Experiment
		5.6 Radiative Processes in Astrophysics Part II: The Relativistic Case
			5.6.1 Relativistic Coulomb Scattering or Ionization Losses
				5.6.1.1 Ionization Loss
				5.6.1.2 Coulomb Scattering
			5.6.2 Inverse Compton Scattering
			5.6.3 Synchrotron Radiation
				5.6.3.1 From the Observer Point of View
				5.6.3.2 From the Particle Point of View
			5.6.4 Relativistic Bremsstrahlung
			5.6.5 Energy Losses: Summary
		5.7 Ultrahigh Energetic (UHE) Processes
			5.7.1 Cosmic Ray Case
			5.7.2 Photons and Neutrinos Cases
		5.8 Indirect Detection of Gamma Ray
			5.8.1 The Principle
			5.8.2 Galactic Halo
			5.8.3 Adiabatic Compression Mechanism
		5.9 The Tricky Case of the Galactic Center
			5.9.1 The Idea
			5.9.2 Dark Matter Density Profiles
			5.9.3 Gamma-Ray Flux from Dark Matter Annihilation
				5.9.3.1 Prompt Gamma Rays
				5.9.3.2 Gamma Rays from Inverse Compton Scattering
		5.10 Dark Matter and Synchrotron Radiation
			5.10.1 Neglecting Diffusion
			5.10.2 Synchrotron Loss of Energy
			5.10.3 Taking into Account Spatial Diffusion**
			5.10.4 General Astrophysical Setup*
				5.10.4.1 Astrophysical Uncertainties
				5.10.4.2 Synchrotron Signal for Different Choices of DM Density Profile
				5.10.4.3 Synchrotron Signal for Different Choices of Cosmic Ray Parameters
				5.10.4.4 Synchrotron Signal for Different Choices of Magnetic Field**
		5.11 Sommerfeld Enhancement
			5.11.1 Generalities
			5.11.2 Solving Schrodinger\'s Equation
			5.11.3 The Coulomb Potential
			5.11.4 The Yukawa Interaction
		5.12 Structure Formation Constraints
			5.12.1 Free Streaming
			5.12.2 Jeans Radius and Mass
			5.12.3 The Influence of Dark Matter
			5.12.4 Correlation Function*
			5.12.5 Power Spectrum P(k)*
		References
Appendix
A Cosmology and Astrophysics
	A.1 Useful Cosmology
		A.1.1 Lorentz Transformation
		A.1.2 Friedmann Equation
		A.1.3 The Horizon
	A.2 Basics of General Relativity
		A.2.1 The Context
		A.2.2 Measuring a Length, a Surface, or a Volume
		A.2.3 The Einstein-Hilbert Action (I)
		A.2.4 Tooling with the Metric
		A.2.5 The Geodesic Equation
		A.2.6 A Geometrical Approach
		A.2.7 The Riemann Tensor
		A.2.8 The Einstein Equation of Fields in Vacuum
		A.2.9 Adding Matter Fields
		A.2.10 The Perfect Fluid Stress-Energy-Momentum Tensor
		A.2.11 The Schwarzschild Solution
		A.2.12 The de Sitter Solution
		A.2.13 Metric Solutions as Moving Medium
		A.2.14 Conformal Transformation
		A.2.15 The Graviton
			A.2.15.1 The Canonical Field
			A.2.15.2 Fixing the Gauge
			A.2.15.3 The Lagrangian
		A.2.16 Deflection Angle
	A.3 Matter/Radiation Domination
	A.4 Thermodynamical Fundamental Relations
	A.5 Classical Thermodynamics: The Laplace Law
	A.6 Tooling with Math
		A.6.1 Function (z)
			A.6.1.1 Definition, Propriety
			A.6.1.2 Some Values
		A.6.2 The Riemann Zeta Function ζ(z)
			A.6.2.1 Definition, Propriety
			A.6.2.2 Some Values
		A.6.3 Modified Bessel Function of the Second Kind Kn(z)
			A.6.3.1 Definition, Propriety
			A.6.3.2 Some Values
			A.6.3.3 Some Approximations
		A.6.4 Useful Integrals
			A.6.4.1 Euler-Mascheroni Constant γ
			A.6.4.2 Gauss Error Function erf
			A.6.4.3 Delta Dirac δ
		A.6.5 Laplace Operator
	Reference
B Particle Physics
	B.1 Feynman Rules
		B.1.1 Decay Rates and Cross Sections
	B.2 Feynman Rules
		B.2.1 General Rules
		B.2.2 Majorana Rules
			B.2.2.1 Another Interpretation
		B.2.3 Standard Model Couplings
	B.3 Diracology
		B.3.1 Matrices
		B.3.2 Dirac Equation
		B.3.3 The Spin Matrix
		B.3.4 Proca Equation
		B.3.5 Rarita-Schwinger Equation
		B.3.6 Parity Operator
			B.3.6.1 Fermion Case
			B.3.6.2 Boson Case
		B.3.7 The Charge Conjugate Operator
		B.3.8 The Majorana Case
			B.3.8.1 Definition
			B.3.8.2 Dirac-Like Majorana Equation
			B.3.8.3 In the ``Left-Right\'\' Representation
			B.3.8.4 Furry\'s Theorem
		B.3.9 Traces
		B.3.10 Mandelstam Variables
		B.3.11 The Generators Tai
	B.4 Lorentz Invariant Scattering Cross Section and Phase Space
		B.4.1 Fermi\'s Golden Rule
			B.4.1.1 The Non-relativistic Case
			B.4.1.2 Normalization
			B.4.1.3 The S-Operator
			B.4.1.4 Computing the Rate
			B.4.1.5 Application
		B.4.2 Special Case
		B.4.3 Computing the Phase Space
			B.4.3.1 Two-Body Phase Space
			B.4.3.2 N-Body Phase Space
			B.4.3.3 Summary in the Massless Case
			B.4.3.4 Examples
		B.4.4 Annihilation
			B.4.4.1 General Formulae
			B.4.4.2 A Shorter Formulation
			B.4.4.3 A Note on the Symmetry Factor
			B.4.4.4 The Specific Case of Majorana or Identical Initial Particle
			B.4.4.5 Unitarity Limit
			B.4.4.6 Scalar Fermi-Like Interaction
			B.4.4.7 Vector Fermi-Like Interaction
			B.4.4.8 Neutrino Interaction
			B.4.4.9 Annihilation into Monochromatic Photons
			B.4.4.10 Annihilation in the Case of Real Scalar Dark Matter to Pairs of Fermions
			B.4.4.11 Annihilation in the Case of Vectorial Dark Matter to Pairs of Fermions
			B.4.4.12 Exchange of a Vector: The Case of the Vectorial Coupling
		B.4.5 Spin-Independent Diffusion, Elastic Scattering
		B.4.6 Decaying Particles
			B.4.6.1 Two-Body Decay
			B.4.6.2 Three-Body Decay
			B.4.6.3 Application to Muon Decay
		B.4.7 Higgs Lifetime
			B.4.7.1 Higgs Lifetime from H →f (p1)  (p2)
			B.4.7.2 Higgs Lifetime from H →Z(p1)Z(p2)
			B.4.7.3 Higgs Lifetime from H →W+W-
			B.4.7.4 Higgs Lifetime from H →SS with Singlet Scalar Dark Matter
			B.4.7.5 Higgs Lifetime from H →XμXμ
			B.4.7.6 General Scalar Width: T →χ
		B.4.8 Majorana Case
		B.4.9 Vector Lifetime
			B.4.9.1 Generalities
			B.4.9.2 W and Z Widths
	B.5 s-Wave, p-Wave, Helicity Suppression, and All that
		B.5.1 Velocity Suppression
		B.5.2 Spin Selection
		B.5.3 Application to Specific Models
		B.5.4 Helicity Suppression
		B.5.5 Summary
	B.6 Schrodinger Equation
		B.6.1 Generalities
		B.6.2 Solutions
	B.7 The Strong-CP Problem
		B.7.1 QCD Lagrangian
		B.7.2 The Axionic Peccei-Quinn Solution
	B.8 Useful Spectrum
		B.8.1 Gamma Spectrum
		B.8.2 Positron Spectrum
		B.8.3 Antiproton Spectrum
	B.9 Lagrangians
		B.9.1 Standard Model
			B.9.1.1 Higgs Couplings
			B.9.1.2 Vectorial Couplings
		B.9.2 Singlet Scalar
		B.9.3 Extra U(1) and Kinetic Mixing
	B.10 Unification
		B.10.1 A Brief Review
		B.10.2 SU(5): The Generators
		B.10.3 SU(5): The Spinor Representation
		B.10.4 Proton Decay
		B.10.5 Breaking of SU(5)
		B.10.6 Doublet-Triplet Splitting
		B.10.7 SO(10) and Unification of Coupling Constants*
		B.10.8 Computing β-Functions
	References
C Neutrino Physics
	C.1 Astrophysical and Cosmological Sources of Neutrino
		C.1.1 Solar Neutrinos
		C.1.2 Atmospheric Neutrinos
	C.2 Ultra High Energetic Neutrinos
	C.3 Neutrino Mass
		C.3.1 Dirac Mass
		C.3.2 Majorana Mass
			C.3.2.1 Without Right-Handed Neutrino
			C.3.2.2 With Right-Handed Neutrino
	C.4 The See-Saw Mechanism
		C.4.1 A Simple Example
		C.4.2 Generalization
		C.4.3 The Specific Case mML=0
			C.4.3.1 The Eigenvalues
			C.4.3.2 The Eigenvectors
		C.4.4 An Application: Coupling to a Scalar Field (Majoron)
D Useful Statistics
	D.1 5 σ and p-Value
		D.1.1 5σ
		D.1.2 p-Value
	D.2 Systematics vs Statistics
	D.3 Look-Elsewhere Effect (LEE)
		D.3.1 Generality
		D.3.2 Applying the LEE Effect to the Higgs Discovery
		D.3.3 Applying the LEE Effect to the Dark Matter Searches
	D.4 Bayesian vs Frequentist Approach
E Numbers
	E.1 Useful Formulae
		E.1.1 Cosmology
		E.1.2 Particle Physics
	E.2 Tables
Index