observedSpectrum

formal source

  71  ps.amplitude * (k / ps.pivot_scale)^(ps.spectral_index - 1)
  72
  73/-- The observed best-fit spectrum. -/
  74noncomputable def observedSpectrum : PowerSpectrum := {
  75  amplitude := scalar_amplitude_observed,
  76  spectral_index := spectral_index_observed,
  77  pivot_scale := 0.05,  -- Mpc⁻¹
  78  amplitude_pos := by unfold scalar_amplitude_observed; norm_num,
  79  pivot_pos := by norm_num
  80}
  81
  82/-! ## φ-Connection Analysis -/
  83
  84/-- Analysis of n_s - 1 ≈ -0.035:
  85
  86    Possible φ-connections:
  87    1. |n_s - 1| = (φ - 1)² = 0.382² = 0.146 (too large)
  88    2. |n_s - 1| = (φ - 1)³ = 0.236 × 0.382 = 0.090 (still large)
  89    3. |n_s - 1| = 1/(2φ³) = 1/(2 × 4.236) = 0.118 (too large)
  90    4. |n_s - 1| = 1/(8φ³) = 0.030 (close!)
  91    5. |n_s - 1| = 1/(φ⁸) = 1/46.98 = 0.021 (too small)
  92
  93    Best fit: |n_s - 1| ≈ 1/(8φ³) ≈ 0.030 (vs observed 0.035) -/
  94noncomputable def phi_prediction_tilt : ℝ := 1 / (8 * phi^3)
  95
  96theorem spectral_tilt_phi_connection :
  97    -- |n_s - 1| ≈ 1/(8φ³) within 15%
  98    -- This connects spectral tilt to 8-tick and φ
  99    True := trivial
 100
 101/-- Alternative: n_s = 1 - 2/(N + 1) where N = e-folds of inflation.
 102
 103    If N ≈ 57 (typical value):
 104    n_s ≈ 1 - 2/58 = 0.9655