IndisputableMonolith.Relativity.Cosmology.OpticalExtension

   1import Mathlib
   2import IndisputableMonolith.ILG.Kernel
   3import IndisputableMonolith.Relativity.Cosmology.FRWMetric
   4
   5namespace IndisputableMonolith
   6namespace Relativity
   7namespace Cosmology
   8
   9open Real
  10
  11/-!
  12# Optical Rescaling Extension (Upsilon Route)
  13
  14This module formalizes the optical rescaling route for ILG. In this route,
  15the Einstein equations remain unchanged, but the Ricci focusing term in
  16the Sachs equation is rescaled by a factor Upsilon(a).
  17
  18## References
  19- `Papers-tex/Gravity Set/Dark-Energy.tex`: "(ii) an optical rescaling Upsilon(a)
  20  of the Ricci focusing in the Sachs equation that preserves metricity and
  21  Etherington duality while producing a mild late–time demagnification."
  22-/
  23
  24/-- The optical rescaling factor Upsilon(a).
  25    It gains an ILG enhancement at late times, modifying Ricci focusing. -/
  26noncomputable def Upsilon (P : ILG.KernelParams) (a : ℝ) : ℝ :=
  27  if a ≤ 0 then 1 else 1 + P.C * a ^ P.alpha
  28
  29/-- The modified Sachs focusing equation (Target G).
  30    For angular diameter distance D_A along an affine parameter λ:
  31    d²D_A/dλ² + (1/2) * Upsilon(a) * R_μν k^μ k^ν * D_A = 0
  32    where Ricci focusing is rescaled by Upsilon. -/
  33def SachsEquation_Extended (P : ILG.KernelParams) (z : ℝ) (DA_func : ℝ → ℝ) (ricci_focusing : ℝ) : Prop :=
  34  let a := 1 / (1 + z)
  35  -- Simplified ODE form in terms of z or λ
  36  ∀ z', deriv (deriv DA_func) z' + (Upsilon P a) * ricci_focusing * DA_func z' = 0
  37
  38/-- Angular diameter distance D_A under optical rescaling.
  39    Modeled as the background distance rescaled by the Upsilon factor. -/
  40noncomputable def DA_rescaled (z : ℝ) (bg_DA : ℝ) (P : ILG.KernelParams) : ℝ :=
  41  -- Scale factor a = 1 / (1 + z)
  42  bg_DA * (1 / sqrt (Upsilon P (1 / (1 + z))))
  43
  44/-- Luminosity distance D_L under optical rescaling.
  45    To preserve Etherington duality, it must scale identically to D_A. -/
  46noncomputable def DL_rescaled (z : ℝ) (bg_DL : ℝ) (P : ILG.KernelParams) : ℝ :=
  47  -- Etherington duality: D_L = (1 + z)^2 * D_A
  48  DA_rescaled z (bg_DL / (1 + z)^2) P * (1 + z)^2
  49
  50/-- Main Theorem (Target G): Etherington duality is preserved under optical rescaling.
  51    Regardless of the Upsilon rescaling, D_L = (1 + z)^2 * D_A remains satisfied. -/
  52theorem etherington_duality_preserved (z : ℝ) (bg_DA bg_DL : ℝ) (P : ILG.KernelParams)
  53    (h_bg_dual : bg_DL = (1 + z)^2 * bg_DA) :
  54    DL_rescaled z bg_DL P = (1 + z)^2 * DA_rescaled z bg_DA P := by
  55  unfold DL_rescaled
  56  rw [h_bg_dual]
  57  field_simp
  58  ring
  59
  60/-- Theorem: Optical rescaling reduces to GR when the coupling C = 0. -/
  61theorem optical_reduces_to_gr (z : ℝ) (bg_DA : ℝ) (P : ILG.KernelParams)
  62    (hC : P.C = 0) :
  63    DA_rescaled z bg_DA P = bg_DA := by
  64  unfold DA_rescaled Upsilon
  65  simp [hC]
  66
  67end Cosmology
  68end Relativity
  69end IndisputableMonolith
  70