18-04 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 dq ² + r ² sin²q df²],
    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 = - (4pG/3)(r + 3p/c²)R + Lc²R/3
    
    (dR/dt)² + kc² = 8pGrR²/3 + Lc²R²/3

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    Recall that the Hubble parameter H  is (dR/dt)/R, then
    1 = (8pG/3H²) r + (c²/3H²) L - (c²/R²H²) k
        = (8pG/3H²) [r + (c²/8pG) L - (3c²/8pGR²) k]
        = W_m + W_L + W_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
    r_c = 3H² / 8pG = 1.8743 x 10^-29 h² g/cm³
    (This is so defined by requiring k=0 in the Einstein's field equation with the Robertson-Walker metric.)
    r_c = 1.0 x 10^-29 g/cm³ , for h = 0.73
     
    for density < r_c , the universe is open (negative curvature, k < 0, hyperbolic)
    for density = r_c , the universe is flat (zero curvature, k=0)
    for density > r_c , the universe is closed (positive curvature, k > 0, spherical)
    
    [geometry of the universe ]
     
    (recall that the density r_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?
    r₀ = r_m+r_L (r_L is the density of dark energy)
    W_m = r_m/r_c
    W_L = r_L/r_c
    W₀ = W_m+W_L
    A flat universe means W₀ = 1
     
• Check the expansion history using SN Ia
  
  [Varying rate of cosmic expansion ]
  
  [Hubble diagram of high-z SN Ia ]
  
  [Constraints on cosmological parameters]
  
  The expansion of our universe is accelerating!
  What is exactly the 'dark energy'? Cosmological constant L? Quintessence? Or?
  
  [Cosmic energy density evolution]
  
   
• Growth of the universe
  The 'deceleration parameter' q is defined as
  -R(d²R/dt²)/(dR/dt)²   =   -(d²R/dt²)/(H²R),
  which, when radiation can be ignored, can be written as
  q₀ = 0.5 W_m - W_L  , in the Friedmann-Robertson-Walker (FRW) model with the cosmological constant.
   
  With a determined k (therefore W_m + W_L is known; now it seems that k=0), the value of q₀ tells us how the universe evolves.
  
  [The growth of our universe, assumed being 'flat'. ]