The Friedmann equations are the backbone of modern cosmology, describing how the universe expands over time. These equations, derived from Einstein's general relativity, link the universe's expansion rate to its energy density and curvature.

Cosmic expansion, first observed by Edwin Hubble, is now known to be accelerating due to mysterious dark energy. This discovery has profound implications for our understanding of the universe's past, present, and future, challenging previous assumptions about its evolution.

Friedmann Equations and Cosmological Models

Fundamental Equations and Parameters

• Friedmann equations describe the expansion of the universe derived from Einstein's field equations in general relativity
• First Friedmann equation relates the expansion rate to the energy density and curvature of the universe: $(\frac{\dot{a}}{a})^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2}$
• Second Friedmann equation describes the acceleration of the expansion: $\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}(\rho + \frac{3p}{c^2})$
• Scale factor a(t) represents the relative size of the universe at time t compared to its current size
  • Directly related to the redshift of distant objects
  • Evolves according to the Friedmann equations
• Einstein-de Sitter model assumes a flat universe dominated by matter
  • Predicts a deceleration in the expansion rate over time
  • Scale factor in this model evolves as $a(t) \propto t^{2/3}$

Density Concepts and Cosmic Parameters

• Critical density defines the boundary between an open and closed universe
  • Calculated using the Hubble parameter: $\rho_c = \frac{3H^2}{8\pi G}$
• Density parameter Ω compares the actual density of the universe to the critical density
  • Ω < 1 indicates an open universe (negative curvature)
  • Ω = 1 represents a flat universe (zero curvature)
  • Ω > 1 suggests a closed universe (positive curvature)
• Current observations indicate Ω ≈ 1, suggesting a nearly flat universe
• Total density parameter includes contributions from matter, radiation, and dark energy: $\Omega_{total} = \Omega_m + \Omega_r + \Omega_Λ$

Cosmic Expansion and Acceleration

Expansion Dynamics and Observational Evidence

• Cosmic expansion describes the ongoing increase in distance between all points in the universe
  • First observed by Edwin Hubble through the redshift of distant galaxies
  • Characterized by Hubble's Law: v = H₀d, where H₀ is the Hubble constant
• Acceleration of the universe discovered in 1998 through observations of Type Ia supernovae
  • Contradicted previous assumptions of a decelerating expansion
  • Implies the existence of a repulsive force counteracting gravity at large scales
• Observational techniques for studying cosmic expansion include:
  • Measuring the redshift of distant galaxies
  • Analyzing the cosmic microwave background radiation
  • Studying baryon acoustic oscillations in the large-scale structure of the universe

Dark Energy and Cosmological Implications

• Cosmological constant Λ introduced by Einstein to represent a constant energy density of space
  • Initially proposed to create a static universe model
  • Later repurposed to explain the accelerating expansion
• Dark energy serves as the leading explanation for cosmic acceleration
  • Comprises about 68% of the energy content of the universe
  • Exhibits negative pressure, causing the expansion to accelerate
• Potential forms of dark energy include:
  • Vacuum energy from quantum field theory
  • Scalar fields (quintessence models)
  • Modified gravity theories
• Implications of cosmic acceleration for the fate of the universe:
  • Continued acceleration could lead to a "Big Rip" scenario
  • Possibility of dark energy density changing over time (dynamic dark energy models)
  • Challenges for structure formation in the far future as galaxies become increasingly isolated