The Early Universe

What this course covers

• Standard Big Bang model overview
• Limits of the conventional Big Bang
• Cosmic inflation as a prequel to the Big Bang
• Gravitational repulsion and negative pressure
• Energy conservation during inflation
• Large-scale uniformity and the horizon problem
• Flatness problem and omega
• Origin of density fluctuations from quantum effects
• Eternal inflation and the multiverse
• Dark energy and vacuum energy
• String theory landscape and the cosmological constant problem
• Standard Big Bang and cosmic inflation overview
• Evidence for cosmic inflation
• Eternal inflation and the multiverse
• Dark energy and the vacuum energy problem
• String theory landscape of vacua
• Anthropic selection effect argument
• Hubble's law
• Measuring galactic distances with standard candles
• Doppler shift: framework and key quantities
• Doppler shift: stationary observer, moving source
• Redshift parameter z
• Doppler shift: moving observer, stationary source
• Low-velocity equivalence of the two Doppler cases
• Origin and postulates of special relativity
• Time dilation
• Relativistic Doppler shift derivation
• Ideal clocks and acceleration in special relativity
• Lorentz contraction
• Relativity of simultaneity
• Isotropy of the universe
• Cosmic Microwave Background (CMB)
• Homogeneity of the universe
• Hubble's Law and the Hubble constant
• Hubble expansion as uniform, observer-independent expansion
• Comoving coordinates and the scale factor
• Deriving Hubble's Law from the scale factor
• Light ray propagation in comoving coordinates
• Cosmic time and clock synchronization
• Comoving coordinates and cosmic time synchronization
• Cosmological redshift derivation
• Cosmological vs. special-relativistic redshift
• Scale factor linear growth in absence of gravity
• Dynamics of homogeneous expansion: role of gravity
• Newton's static infinite universe and its flaw
• Gauss's law for gravity
• Gravitational potential and Poisson's equation
• Absolute vs. conditional convergence of integrals
• Ambiguity of gravitational force in infinite mass distributions
• Instability of a homogeneous infinite universe
• Ambiguity of action-at-a-distance gravity
• Symmetry argument and relative acceleration
• Building a Newtonian model universe
• Shell model and equations of motion
• Shell crossing and validity of equations
• Scale factor and preservation of homogeneity
• Friedmann-like equation from energy conservation
• Homogeneous expanding universe model setup
• Newton's shell theorem for gravity
• Shell tracking function and no-shell-crossing argument
• Equation of motion for a shell and scale factor
• Friedmann equations derivation
• Comoving coordinates and units (notches)
• Open, closed, and flat universes
• Critical density and Omega parameter
• Scale factor evolution for flat universe (a ∝ t²/³)
• Newtonian model of an expanding universe
• Friedmann equations
• Evolution of matter density with scale factor
• Co-moving coordinates and the notch unit
• Open, closed, and flat universe solutions
• Critical density and the Omega parameter
• Scale factor evolution in a flat matter-dominated universe
• Age of a flat matter-dominated universe
• Meaning and limits of the Big Bang singularity
• Horizon distance
• Parametric solution for closed universe
• Age of a flat matter-dominated universe
• Big Bang singularity and its physical limits
• Cosmological horizon distance
• Friedmann equation for a closed universe
• Parametric (cycloid) solution for a closed universe
• Cycloid geometry of closed-universe evolution
• Role of parameter α and the development angle θ
• Age of a closed universe in terms of H and Ω
• Parametric evolution of a closed universe
• Age of a closed universe in terms of H and Ω
• Evolution and age of an open universe
• Scale factor evolution curves for all three universe types
• Scope of general relativity in this course
• Euclid's five postulates and the fifth postulate problem
• Gauss–Bolyai–Lobachevsky (GBL) non-Euclidean geometry
• Inner vs. outer properties of curved surfaces
• The metric as a quadratic distance function
• Gravity as spacetime curvature in general relativity
• Euclid's fifth postulate and non-Euclidean geometry
• Metric (distance function) description of geometry
• Local flatness of curved spaces
• Metric of a 2-D spherical surface
• 3-D spherical surface in 4-D Euclidean space
• Homogeneity and isotropy of the closed universe
• Alternative radial coordinates (u and r̃)
• Connecting spatial curvature to the Friedmann equation
• Robertson–Walker metric
• Metric of a two-dimensional sphere
• Three-dimensional closed universe metric
• Orthogonality of displacement vectors on the 3-sphere
• Robertson-Walker metric and its derivation
• Open and flat universe metrics (negative and zero k)
• Homogeneity of the Robertson-Walker metric
• Finite vs. infinite size of universe models
• Spacetime interval and Lorentz invariance
• Robertson-Walker metric and homogeneity
• Universality of the Robertson-Walker metric
• Lorentz-invariant spacetime interval
• Coordinates as arbitrary labels in GR
• Equivalence principle and free-falling observers
• Robertson-Walker spacetime metric
• Einstein summation convention and metric notation
• Geodesic as a path of stationary length
• Deriving the geodesic equation
• Robertson-Walker spacetime metric
• Parametrizing paths and arc length
• Calculus of variations for geodesics
• Integration by parts in the geodesic derivation
• Geodesic equation and arc-length parametrization
• Christoffel symbols and second-derivative form
• Schwarzschild metric and Schwarzschild radius
• Radial free-fall geodesic in Schwarzschild spacetime
• Coordinate time vs. proper time at the Schwarzschild horizon
• Spacetime geodesic equation review
• E = mc² and mass-energy equivalence
• Relativistic energy-momentum four-vector
• Relativistic mass and gravitational sourcing
• Photon energy density and redshift scaling
• Matter-radiation equality and radiation-dominated era
• Friedmann equations for matter-dominated universe
• Radiation pressure and modified fluid equation
• Modified Friedmann equation for radiation (pressure term)
• Relativistic energy-momentum four-vector
• Radiation energy density and redshifting
• Matter-radiation equality epoch
• Pressure in the Friedmann equations
• Pressure as a source of gravity and dark energy
• Radiation-dominated flat universe dynamics
• Black-body (cavity) radiation
• Jeans paradox and quantum resolution
• Black-body thermodynamic formulas
• Neutrino flavors, masses, and cosmological role
• Flat radiation-dominated universe dynamics
• Black-body radiation thermodynamics
• Entropy conservation in the early universe
• Fermions vs. bosons and spin-statistics
• Neutrino properties and helicity
• Effective degrees of freedom g across particle thresholds
• Neutrino temperature after electron-positron annihilation
• Dirac vs. Majorana neutrino mass
• Temperature as a function of cosmic time
• Recombination and photon decoupling
• Black-body radiation formulas and particle types
• Neutrino temperature vs. photon temperature after e+e- annihilation
• Neutrino mass and helicity: Dirac vs. Majorana
• CMB spectrum and black-body radiation confirmation
• Black-body spectral energy density formula
• Discovery of cosmic acceleration via Type Ia supernovae
• Cosmological constant and vacuum energy density
• Equation of state of vacuum energy (negative pressure)
• Friedmann equations with cosmological constant
• Exponential expansion dominated by vacuum energy
• Effect of vacuum energy on the age of the universe
• Cosmic microwave background measurement history
• Cosmological constant and vacuum energy
• Gravitational effect of pressure
• Friedmann equation with vacuum energy
• Omega parameters and critical density
• Age of the universe integral formula
• Analytic age solutions with matter and vacuum energy
• Robertson-Walker metric and comoving coordinates
• Luminosity-redshift relation for standard candles
• Comoving distance integral and supernova evidence for dark energy
• Age of the universe calculation
• Radiation flux versus redshift formula
• Supernova type 1a observations and the distance modulus
• Why dimmer supernovae imply cosmic acceleration
• Systematic error arguments: dust and galactic evolution
• Quantum zero-point energy of bosonic fields
• Negative vacuum energy contribution from fermionic fields
• Nonzero vacuum field values and the hierarchy problem
• Einstein–Friedmann historical dispute
• Horizon homogeneity problem
• Observational evidence for accelerating universe
• Vacuum energy and cosmological constant
• Anthropic selection of vacuum energy
• Horizon (homogeneity) problem
• Flatness problem
• Standard Model of particle physics
• Gauge theories and non-Abelian gauge symmetry
• Grand unified theories and SU(5)
• Standard Model gauge symmetry groups
• U(1) as the gauge symmetry of electromagnetism
• Non-abelian gauge symmetries of weak and strong forces
• Electroweak unification and symmetry breaking
• Spontaneous symmetry breaking mechanism
• Higgs field and particle mass generation
• Topological defects from symmetry breaking domains
• Magnetic monopoles as point-like topological defects
• Monopole properties and the cosmological monopole problem
• Scalar fields and false vacuum
• Negative pressure of the false vacuum
• Gravitational repulsion and exponential expansion
• New inflationary scenario and graceful exit
• Initial conditions and origin of inflation
• Cosmological no-hair theorem and de Sitter space
• Event horizons in de Sitter space
• Sample numbers for GUT-scale inflation
• Resolution of the horizon/homogeneity problem
• Resolution of the flatness problem
• Resolution of the monopole problem
• Quantum fluctuations and CMB power spectrum

Syllabus

1. Lectures

   1. 1. Inflationary Cosmology: Is Our Universe Part of a Multiverse? Part I
   2. 2. Inflationary Cosmology: Is Our Universe Part of a Multiverse, Part II
   3. 3. The Doppler Effect and Special Relativity
   4. 4. The Kinematics of the Homogeneous Expanding Universe
   5. 5. Cosmological Redshift and the Dynamics of Homogeneous Expansion, Part I
   6. 6. The Dynamics of Homogeneous Expansion, Part II
   7. 7. The Dynamics of Homogeneous Expansion, Part III
   8. 8. The Dynamics of Homogeneous Expansion, Part IV
   9. 9. The Dynamics of Homogeneous Expansion, Part V
   10. 10. Introduction to Non-Euclidean Spaces
   11. 11. Non-Euclidean Spaces: Closed Universes
   12. 12. Non-Euclidean Spaces: Open Universes and the Spacetime Metric
   13. 13. Non-Euclidean Spaces: Spacetime Metric and Geodesic Equation
   14. 14. The Geodesic Equation
   15. 15. Black-Body Radiation and the Early History of the Universe, Part I
   16. 16. Black-Body Radiation and the Early History of the Universe, Part II
   17. 17. Black-Body Radiation and the Early History of the Universe, Part III
   18. 18. Cosmic Microwave Background Spectrum and the Cosmological Constant, Part I
   19. 19. The Cosmological Constant, Part II
   20. 20. Supernovae Ia and Vacuum Energy Density
   21. 21. Problems of the Conventional (Non-inflationary) Hot Big Bang Model
   22. 22. The Higgs Field and the Cosmological Magnetic Monopole Problem
   23. 23. Inflation