17-01 The Expanding Universe

• Around Newton's time (1642 ~ 1727) in the western world, people believed that the universe is static and infinite.
  But then, why is the sky dark in the night? -- Olbers's paradox
  
  
  [Olbers's paradox: the night sky should be as bright as a star's surface if the universe is static and infinite. (Brightness in a unit area in the sky remains the same no matter how far away stars are.)]
  
   
• Cosmology is the science of exploring the origin and evolution of the universe.
   
• The Hubble expansion: more distant galaxies have larger redshifts.
  
  Are we at the center of the expansion?
  
  
• The Cosmological principle
  
  Any observer in any galaxy sees the same general features of the universe.
  It is an extension of the Copernican principle, which says simply we are not privileged observers.
  ==> The assumptions of homogeneity and isotropy
   
  
• The Hubble expansion is not simply objects flying apart. It is a global expansion of space.
  
  
  
   
• Cosmological redshifts
  -- caused by the expansion of space
  
  
  
  z = (λ_r - λ_e) / λ_e
   
  1 + z = λ_r/λ_e, which is the ratio of the current value of a typical cosmological scale to that at the time of emission.
   
• *** ADVANCED MATERIALS ***

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  The Robertson-Walker metric
  A description of  the dynamics of the universe should be based on general relativity.
  The most general metric satisfying the cosmological principle is the Robertson-Walker metric:

  ds² = c²dt² - R²(t) [(1-kr²)^-1 dr²  + r² dθ ² + r ² sin²θ dφ²]

  where R(t) is a time-dependent scale factor of the universe, k (= +1, 0, or -1) is related to the curvature of space, and r is a time-independent comoving coordinate.
  Then Einstein's field equations give the evolution of the scale factor.
   
  The Hubble parameter H  is (dR/dt)/R, and its current value is denoted as H₀.

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• H  is in general a function of time.
  Because H  is difficult to determine, and more importantly, z = R(t₀)/R(t) - 1,
  astronomers use simply the redshift z to refer to the distant past.
   
• Cosmologists usually insert this metric form into Einstein's field equations, and study the behavior of the scale factor R(t) to describe the dynamics of the cosmic expansion.
   
• In order to have a static solution -- at the time before Hubble discovered the cosmic expansion -- Einstein added one more term, called cosmological constant Λ, to his field equations.
   
• *** ADVANCED MATERIALS ***

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  Einstein's field equations
  
  G^μν + Λ g^μν = - (8πG/c⁴) T^μν
  
  G^μν = R^μν - (1/2) g^μν R
  
  g^μν is the metric tensor.
  R^μν is the Ricci tensor and R the curvature scalar. They both are contraction of the Riemann tensor, which consists of the metric up to the second derivative. (Do not confuse with the scale factor R(t) in the Robertson-Walker metric.)
  T^μν is the density tensor of matter distribution.

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