Friedmann Equations

Cosmology · Part 1

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Table of Contents

  1. 1.1. The Cosmological Principle
  2. 2.2. Constructing the FLRW Metric from Symmetry
  3. 3.3. Metric Tensor Components
  4. 4.4. Christoffel Symbols — Full Computation
  5. 5.5. The Ricci Tensor
  6. 6.6. Einstein's Field Equations
  7. 7.7. Deriving the Friedmann Equations
  8. 8.8. The Continuity Equation
  9. 9.9. Equation of State and Solutions for Each Component
  10. 10.10. Density Parameters and the Friedmann Constraint

Key Equations

$$N_{\text{max}} = \frac{d(d+1)}{2}$$
$$\Gamma^0{}_{11} = \frac{a\dot{a}}{c^2(1-kr^2)}, \qquad \Gamma^0{}_{22} = \frac{a\dot{a}\,r^2}{c^2}, \qquad \Gamma^0{}_{33} = \frac{a\dot{a}\,r^2\sin^2\theta}{c^2}$$
$$T_{00} = \rho c^2, \qquad T_{0i} = 0$$
$$\rho_c \equiv \frac{3H^2}{8\pi G} \approx 1.88 \times 10^{-29}\,h^2\;\text{g/cm}^3$$

Equations are rendered with MathJax in the PDF with professional LaTeX typesetting.

Course Context

This PDF is part of the Cosmology course on CoursesHub.World. Study the origin, structure, and evolution of the universe. Covers the expanding universe, thermal history, cosmic inflation, CMB physics, structure formation, dark matter, dark energy, and observatio...

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