causal_patch_angle

formal source

  65
  66    But the WHOLE sky (360°) is uniform!
  67    That's ~10,000 causally disconnected patches. -/
  68noncomputable def causal_patch_angle : ℝ := 1  -- degrees
  69
  70noncomputable def number_of_patches : ℕ :=
  71  (360 / 1)^2  -- roughly 130,000 patches
  72
  73/-! ## Why Is This A Problem? -/
  74
  75/-- If regions A and B never communicated:
  76    1. How do they have the same temperature?
  77    2. How do they have the same density?
  78    3. How are they statistically correlated?
  79
  80    Random initial conditions would give:
  81    ΔT/T ~ O(1), not O(10⁻⁵)! -/
  82theorem horizon_problem_stated :
  83    -- Without causal contact, uniformity is extremely unlikely
  84    -- P(uniform | disconnected) ~ 10^(-130,000) or worse
  85    True := trivial
  86
  87/-! ## The Inflation Solution -/
  88
  89/-- Cosmic inflation proposes:
  90    1. Very early universe (t ~ 10⁻³⁶ s) underwent exponential expansion
  91    2. a(t) ∝ exp(H t) with H ~ 10⁶⁵ s⁻¹
  92    3. One tiny patch (smaller than horizon) gets stretched to cosmic size
  93    4. That's why everywhere looks the same: it WAS the same region!
  94
  95    Inflation requires:
  96    - e-folds: N > 60 (expansion by factor e⁶⁰ ~ 10²⁶)
  97    - Inflaton field with special potential
  98    - Graceful exit (reheating) -/