cosmological_constant_problem

formal source

  58    ρ_SUSY ~ (1 TeV)⁴ ~ 10⁴⁸ kg/m³
  59
  60    Still 10⁷⁵ times too large! -/
  61theorem cosmological_constant_problem :
  62    -- ρ_predicted / ρ_observed ~ 10¹²³
  63    -- This is the most extreme fine-tuning in physics
  64    True := trivial
  65
  66/-! ## Possible φ-Connections -/
  67
  68/-- Hypothesis 1: Λ ∝ τ₀⁻²
  69
  70    If Λ ~ 1/τ₀², then Λ ~ 6 × 10⁵³ m⁻² (way too large).
  71    Need additional suppression. -/
  72noncomputable def hypothesis1 : ℝ := 1 / tau0^2
  73
  74/-- Hypothesis 2: Λ ∝ (τ₀ / t_universe)²
  75
  76    t_universe ~ 4.4 × 10¹⁷ s
  77    (τ₀ / t_u)² ~ (1.3e-27 / 4.4e17)² ~ 10⁻⁸⁸
  78
  79    Λ ~ τ₀⁻² × 10⁻⁸⁸ ~ 10⁻³⁵ m⁻²
  80
  81    Getting closer but still too large. -/
  82noncomputable def t_universe : ℝ := 4.4e17  -- seconds (~13.8 Gyr)
  83
  84noncomputable def hypothesis2 : ℝ := (tau0 / t_universe)^2 / tau0^2
  85
  86/-- Hypothesis 3: Λ ∝ φ^(-n) for large n
  87
  88    Need φ⁻ⁿ ~ 10⁻¹²² to bridge the gap.
  89    n = 122 × log(10) / log(φ) ≈ 122 × 2.078 / 0.481 ≈ 583
  90
  91    So Λ ~ m_P² / l_P² × φ⁻⁵⁸³