17-03 Acceleration of the Cosmic Expansion

• The 'shape' (space geometry) of the universe
   
  • Curved space-time in Einstein's General Relativity
    Recall the Robertson-Walker metric:
    

    ds² = c²dt² - R²(t) [(1-kr²)^-1 dr²  + r² dθ ² + r ² sin²θ dφ²],
    where k is related to the curvature of the spacetime.
     

  • *** ADVANCED MATERIALS ******************************************************************
    

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    Friedmann's cosmological equations: (derived from different components of the field equation)
    
    d²R/dt² = - (4πG/3)(ρ + 3p/c²)R + Λc²R/3
    
    (dR/dt)² + kc² = 8πGρR²/3 + Λc²R²/3

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    Recall that the Hubble parameter H  is (dR/dt)/R, then
    1 = (8πG/3H²)ρ + (c²/3H²)Λ - (c²/R²H²) k
        = (8πG/3H²) [ρ + (c²/8πG)Λ - (3c²/8πGR²) k]
        = Ω_m + Ω_Λ + Ω_k

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  • Depending on the average density, the universe geometry can be open (k < 0), flat (k=0), or closed (k > 0).
    The critical density is
    ρ_c = 3H² / 8πG = 1.8743 x 10^-29 h² g/cm³
    (This is so defined by requiring k = 0 and Λ = 0.)
    ρ_c = 9.0 x 10^-30 g/cm³, for h = 0.7
     
    For density < ρ_c , the universe is open (negative curvature, k < 0, hyperbolic)
    For density = ρ_c , the universe is flat (zero curvature, k=0)
    For density > ρ_c , the universe is closed (positive curvature, k > 0, spherical)
    
    
    [geometry of the universe ]
    
    (Recall that the density ρ_m is only about 2.4 x 10^-30 g/cm³, which includes dark matter in galaxy clusters; do we live in an open universe?)
     
  • From BOOMERANG's measurement of CMB anisotropy, our universe looks flat ! WMAP confirmed that.
    
    
    [CMB and curvature of space; theoretical models of the early universe indicate hot spots in the CMB are typically of one degree in size.]
    
    More dark matter? But not detected with gravity?
    ρ_E = ρ_m+ρ_Λ
    (ρ_Λ is the density of dark energy, whose nature is a big mystery!)
    Ω_m = ρ_m/ρ_c
    Ω_Λ = ρ_Λ/ρ_c
    Ω_E = Ω_m+Ω_Λ
    A flat universe means Ω_E = 1
     
• Check the expansion history using SN Ia
  
  
  [Varying rate of cosmic expansion ]
  
  
  [Hubble diagram of high-z SN Ia: The expansion of our universe is accelerating!]
  
   
• Constraints on cosmological parameters
  
  
  
  
  
  [Cosmic energy density evolution]
  
  What is exactly the 'dark energy'? Cosmological constant Λ? Or something else?
   
• Growth of the universe
  The 'deceleration parameter' q is defined as
  q = - R(d²R/dt²)/(dR/dt)²   =   - (d²R/dt²)/(H²R),
  which, when radiation can be ignored, can be written as
  q = 0.5 Ω_m - Ω_Λ  , in the Friedmann-Robertson-Walker (FRW) model with the cosmological constant.
   
  The value of q₀ tells us how the universe evolves.
  
  
  [The growth of our universe. There could be 'Big Bang' as well as 'Big Crunch'. Now it seems that we have q₀ < 0, if the so-called ΛCDM model is adopted.]