Cosmology Lecture Notes

Cosmology Lecture Notes - Prof. Dr. Matthias Bartelmann

Overview
  • Prof. Dr. Matthias Bartelmann from the Institut für Theoretische Astrophysik, Universität Heidelberg, with contributions from Prof. Dr. Christoph Pfrommer, Leibniz-Institut für Astrophysik Potsdam (AIP), and University of Potsdam.
Contents
  1. The Homogeneous Universe    - 1.1 Geometry and Dynamics       1.1.1 Assumptions       1.1.2 Metric       1.1.3 Cosmological Redshift       1.1.4 Dynamics       1.1.5 Remark on Newtonian Dynamics    - 1.2 Parameters, Age and Distances       1.2.1 Forms of Matter       1.2.2 Parameters       1.2.3 Parameter Values       1.2.4 Age and Expansion of the Universe       1.2.5 Distance Measures       1.2.6 Horizons    - 1.3 Thermal Evolution       1.3.1 Assumptions       1.3.2 Quantum Statistics       1.3.3 Properties of Ideal Quantum Gases       1.3.4 Adiabatic Expansion of Ideal Gases       1.3.5 Particle Freeze-Out    - 1.4 Recombination and Nucleosynthesis       1.4.1 The Neutrino Background       1.4.2 The Entropy of the Universe Today       1.4.3 Photons and Baryons       1.4.4 The Recombination Process       1.4.5 Nucleosynthesis
  2. The Inhomogeneous Universe    - 2.1 The Growth of Perturbations    - 2.2 Statistics and Non-linear Evolution    - 2.3 Evidence for Dark Matter    - 2.4 Spherical Collapse       2.4.1 Collapse of a Homogeneous Overdense Sphere       2.4.2 Connection to Linear Perturbation Theory       2.4.3 Final Density of a Collapsed Halo       2.4.4 The Press-Schechter Mass Function    - 2.5 Halo Formation as a Random Walk    - 2.6 Halo Density Profiles       2.6.1 Isothermal Sphere       2.6.2 Navarro-Frenk-White (NFW) Density Profile
  3. The Early Universe    - 3.1 Cosmological Inflation       3.1.1 Problems       3.1.2 Inflation       3.1.3 Conditions for Inflation       3.1.4 Slow-Roll Conditions       3.1.5 Amount and End of Inflation       3.1.6 Inflation and Structure Formation    - 3.2 Structures in the Cosmic Microwave Background       3.2.1 Simplified Theory of CMB Temperature Fluctuations       3.2.2 CMB Power Spectra and Cosmological Parameters       3.2.3 Foregrounds
  4. The Late Universe    - 4.1 Galaxies and Gas       4.1.1 Ellipticals and Spirals       4.1.2 Spectra, Magnitudes and K-Corrections       4.1.3 Luminosity Functions       4.1.4 Correlation Functions and Biasing       4.1.5 Intervening Gas    - 4.2 Gravitational Lensing       4.2.1 Assumptions, Index of Refraction       4.2.2 Deflection Angle and Lens Equation       4.2.3 Local Lens Mapping and Mass Reconstruction       4.2.4 Deflection by Large-Scale Structures       4.2.5 Limber’s Equation and Weak-Lensing Power Spectra    - 4.3 Galaxy Clusters       4.3.1 Galaxies in Clusters       4.3.2 X-Ray Emission       4.3.3 Gravitational Lensing by Galaxy Clusters       4.3.4 Sunyaev-Zel’dovich Effects       4.3.5 Clusters as Cosmological Tracers    - 4.3.6 Scaling Relations
Chapter 1: The Homogeneous Universe
1.1 Geometry and Dynamics
1.1.1 Assumptions
  • Cosmology is founded on two key assumptions:   1. If the universe is isotropic (uniform in all directions), it must be homogeneous (uniform in all regions).      - Observable properties are isotropic when averaged over sufficiently large scales. Nearby galaxies exhibit anisotropic distribution; further galaxies appear isotropic, and the cosmic microwave background (CMB) is nearly isotropic.   2. The cosmological principle asserts that no observer can claim to be in a preferred position in the Universe, which supports the notion of isotropy and homogeneity.
  • Justifying these assumptions is critical; an ideally homogeneous and isotropic Universe would prevent the existence of structure.   - Among the four fundamental forces, gravity is most relevant for cosmology, described by General Relativity (GR).
1.1.2 Metric
  • The structure of spacetime is described using the metric tensor gννg_{\nu \nu}, which consists of 10 independent components due to symmetry.   - Fundamental observers utilize comoving coordinates, where the line element is simplified under homogeneity and isotropy, leading to the Robertson-Walker metric.
  • The metric is given by:   ds2=−c2dt2+a2(t)(dl2)ds^2 = -c^2dt^2 + a^2(t)(dl^2)   where $a(t)$ is the scaling factor dependent only on time.
1.1.3 Cosmological Redshift
  • Light emitted from a comoving source experiences redshift due to the expanding universe, described by:   - c∣dt∣=a(t)dwc|dt| = a(t)dw
  • This leads to:   - z = rac{a_{o}}{a_{e}} - 1 ightarrow a = rac{1}{1 + z}
1.1.4 Dynamics
  • The dynamics of the universe is captured by Einstein's field equations and remains non-linear, illustrating the challenges in describing metric solutions.
1.1.5 Remark on Newtonian Dynamics
  • Friedmann’s equations can be derived from Newtonian dynamics but the cosmological constant term ($ ext{Λ}$) highlights relativistic effects.
1.2 Parameters, Age and Distances
1.2.1 Forms of Matter
  • Two types of matter are recognized:   1. Relativistic Matter (sources of radiation): pressure approximates P = rac{1}{3} ho c^2.   2. Non-relativistic Matter (dust-like): assumes zero pressure; density decreases as the Universe expands.
1.2.2 Parameters
  • The Hubble parameter ($H$) and associated cosmological parameters are introduced to quantify the expansion of the universe, where $ ho$ denotes density and $ ext{Λ}$ is cosmological constant.   
Chapter 2: The Inhomogeneous Universe
2.1 The Growth of Perturbations
  • Structures such as galaxies stem from perturbations within the cosmic fluid, leading to density variations. Governing equations include continuity and Euler's equations.
2.2 Statistics and Non-linear Evolution
  • Examining density contrast through statistical mechanics enhances understanding of cosmic structure and evolution.
2.3 Evidence for Dark Matter
  • Observational phenomena such as rotation curves demonstrate presence of non-visible mass components in structures.
2.4 Spherical Collapse
  • The collapse of homogeneous overdense spheres provides insight into dark matter halo formation and virialization processes.
Chapter 3: The Early Universe
3.1 Cosmological Inflation
3.1.1 Problems
  • Traditional Big Bang cosmology encounters issues like the horizon and flatness problems, which necessitate the inflationary model.
3.1.2 Inflation
  • Introduces concept of an exponential expansion phase. Conditions under which inflation occurs are analyzed, leading to solutions that address early universe mysteries.
3.2 Structures in Cosmic Microwave Background
  • Explaining temperature fluctuations in CMB helps probe early universe structure formation and its consequences.
Chapter 4: The Late Universe
4.1 Galaxies and Gas
4.1.1 Galaxies in Clusters
  • Examining structures within and around galaxy clusters reveals information about dark matter and cosmological evolution.
4.2 Gravitational Lensing
  • The behavior of light around massive objects, termed gravitational lensing, provides important insights into mass distribution within the universe.
Appendix A: Additional Material
  • Insights into spherical collapse models and alternative methodologies for understanding density contrasts reinforce theoretical frameworks.
Important Concepts and Formulas
  • Friedmann's equations can be summarized as:   extds2extdt2=−GMR2\frac{ ext{d}s^2}{ ext{d}t^2} = -\frac{GM}{R^2}
  • Density scale factors lead to observable phenomena in cosmology, critical for deriving relationships between mass, luminosity, and distance in the universe.