Cosmology Entropy Per Photon Dilution Cubed Eq

The neutrino dilution factor

In the early universe, a plasma of photons, electrons, positrons, and neutrinos shared a common temperature. When the universe cooled enough, electrons and positrons annihilated into photons, dumping their energy into the photon gas. The neutrinos, already decoupled and streaming freely, did not share in this reheating. The ratio of photon temperature to neutrino temperature after this event is a standard piece of cosmology, and it is the cube of this ratio that the declaration dilutionCubed_eq pins down.

The declaration proves, as a theorem, that the quantity dilutionCubed, defined as the ratio of the effective degrees of freedom after annihilation to those before, equals the rational number 4/11. In plain terms, it establishes that the cube of the temperature ratio (T_ν/T_γ)³ is exactly 4/11. This is not a numerical approximation; it is an exact identity derived from the particle content and the assumption of entropy conservation. The proof is a matter of arithmetic: the definition of dilutionCubed unfolds to a ratio of simple integers, and the theorem dilutionCubed_eq verifies that this ratio simplifies to 4/11.

The significance of this exact value is that it feeds directly into the standard calculation of the present-day entropy per photon, a quantity denoted entropyPerPhoton. The framework's library proves that this quantity lies in a narrow window between 7.0393 and 7.0396, a value often rounded to 7.04 in the cosmology literature. The derivation of this window is a chain of theorems: the zeta function ζ(3) is bounded between 1.202042 and 1.202065, π⁴ is bounded between 97.40900 and 97.40914, and the effective number of relativistic degrees of freedom for entropy, gStarS, is proved to be exactly 43/11. The dilutionCubed factor is the key input that makes gStarS equal to 43/11, rather than some other rational number.

What this declaration does not claim is equally important. It does not derive the particle content of the Standard Model; the number of photon polarizations, electron degrees of freedom, and neutrino generations are model inputs, not theorems. It does not prove that entropy is conserved; that is a physical hypothesis imported into the framework. It does not, by itself, establish the entropy-per-photon value; that requires the separate theorems bounding ζ(3) and π⁴. The declaration is a precise, self-contained statement about a ratio of integers, a small but exact piece in a larger cosmological calculation.

theorem dilutionCubed_eq : dilutionCubed = 4 / 11 := by
  unfold dilutionCubed gAfter gBefore gPhoton fermionWeight gElectron
  norm_num

/-- **THEOREM.** `s/n_γ > 7.0393`. -/
theorem entropyPerPhoton_gt : (7.0393 : ℝ) < entropyPerPhoton := by
  have hz := zeta3_lt
  have hp := pi4_gt
  have hzpos := zeta3_pos
  unfold entropyPerPhoton
  rw [lt_div_iff₀ (by linarith : (0 : ℝ) < 45 * zeta3)]
  nlinarith

/-- **THEOREM.** `s/n_γ < 7.0396`. -/
theorem entropyPerPhoton_lt : entropyPerPhoton < (7.0396 : ℝ) := by
  have hz := zeta3_gt
  have hp := pi4_lt
  have hzpos := zeta3_pos
  unfold entropyPerPhoton
  rw [div_lt_iff₀ (by linarith : (0 : ℝ) < 45 * zeta3)]
  nlinarith

/-- **THEOREM.** `g*s = 43/11 ≈ 3.909`. -/
theorem gStarS_eq : gStarS = 43 / 11 := by
  unfold gStarS gPhoton fermionWeight gNeutrino
  rw [dilutionCubed_eq]
  norm_num

What this page does not claim

The declaration does not prove that the Standard Model particle content is correct. The declaration does not establish that entropy is conserved in the early universe. The declaration does not, by itself, derive the value of the entropy per photon.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/EntropyPerPhoton.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: