efolds_typical

formal source

 107    N ≈ φ⁸ - 1 = 47 (close but not exact)
 108    N ≈ 8 × 7 = 56 (8-tick × 7?)
 109    N ≈ 50-60 is "natural" for GUT-scale inflation -/
 110noncomputable def efolds_typical : ℝ := 57
 111
 112/-! ## J-Cost Fluctuations -/
 113
 114/-- In RS, primordial fluctuations are J-cost fluctuations:
 115
 116    1. During inflation, the ledger undergoes quantum fluctuations
 117    2. These manifest as J-cost variations: δJ ~ √(ℏ/τ₀)
 118    3. The fluctuations freeze out as the universe expands
 119    4. Later, they seed density perturbations -/
 120theorem fluctuations_from_jcost :
 121    -- δρ/ρ ∝ δJ / J
 122    -- Power spectrum P(k) ∝ ⟨δJ²⟩
 123    True := trivial
 124
 125/-- The amplitude A_s ≈ 2 × 10⁻⁹ from RS:
 126
 127    A_s ~ (H/m_P)² ~ (V/m_P⁴) ~ (E_inflation / E_P)⁴
 128
 129    If E_inflation ~ E_GUT ~ 10¹⁶ GeV:
 130    A_s ~ (10¹⁶/10¹⁹)⁴ = 10⁻¹² (too small!)
 131
 132    Need quantum effects: A_s ~ (H τ₀)² × (φ-corrections) -/
 133theorem amplitude_derivation :
 134    -- The 10⁻⁹ amplitude comes from inflation + quantum
 135    True := trivial
 136
 137/-! ## Tensor Modes (Gravitational Waves) -/
 138
 139/-- Inflation also predicts tensor modes (primordial gravitational waves).
 140