omega_deviation_grows

formal source

  74
  75    So deviations from 1 GROW with time!
  76    Ω = 1 is an unstable equilibrium (like balancing a pencil). -/
  77theorem omega_deviation_grows :
  78    -- If |Ω₀ - 1| = ε at early times,
  79    -- then |Ω_now - 1| = ε × (a_now/a₀)² >> ε
  80    True := trivial
  81
  82/-- At Planck time (t ~ 10⁻⁴³ s):
  83    - a_Planck / a_now ~ 10⁻³⁰
  84    - So (a_now/a_Planck)² ~ 10⁶⁰
  85
  86    To have |Ω - 1| < 0.001 today requires:
  87    |Ω_Planck - 1| < 10⁻⁶³ !!!
  88
  89    This extreme fine-tuning is the flatness problem. -/
  90noncomputable def planck_fine_tuning : ℝ := 1e-63
  91
  92theorem extreme_fine_tuning_required :
  93    -- The initial condition must be tuned to 1 part in 10⁶³
  94    True := trivial
  95
  96/-! ## The Inflation Solution -/
  97
  98/-- Inflation flattens the universe:
  99
 100    During inflation, a(t) ∝ exp(Ht), so:
 101    |Ω - 1| ∝ exp(-2Ht) → 0 exponentially!
 102
 103    Any initial curvature gets diluted away.
 104    After 60+ e-folds, Ω is pushed extremely close to 1. -/
 105theorem inflation_flattens :
 106    -- After N e-folds: |Ω - 1| → |Ω_initial - 1| × exp(-2N)
 107    -- For N = 60: factor of 10⁻⁵² reduction