Cosmology Neutrino Dilution Dilution Eq Dilution Cubed

Neutrino dilution

In the first second after the Big Bang, the universe was a hot plasma of photons, electrons, positrons, and neutrinos, all in thermal equilibrium. As it expanded and cooled, neutrinos stopped interacting with the other particles and streamed freely. Shortly after, electrons and positrons annihilated into photons, dumping their energy into the photon bath but not into the neutrinos. The photons got hotter; the neutrinos did not. The standard result of cosmology is that the neutrino temperature T_ν ended up lower than the photon temperature T_γ by a factor (4/11)^(1/3), so that (T_ν/T_γ)³ = 4/11.

This ratio is not an arbitrary input. It follows from two physical assumptions: entropy in the coupled photon-electron-positron plasma is conserved during annihilation, and the decoupled neutrinos simply redshift with the expansion. Under those two conditions, the framework's machine-checked library of formal theorems derives the dilution factor. The theorem named dilution_eq_dilutionCubed states that (T_ν/T_γ)³ equals 4/11, matching the value that the framework's entropy ledger (a discrete record of how much entropy each particle species carries) had previously defined by arithmetic. The proof does not assume the 4/11; it computes it from the entropy density of a relativistic plasma, where the 7/8 fermion weight and the 2π²/45 coefficient come from derived integrals, not from fitting.

The same derivation also fixes the present-day effective number of entropy degrees of freedom. With photons at T_γ and six fermionic neutrino states at the cooler T_ν, the total entropy density becomes (2π²/45)·(43/11)·T_γ³. The 43/11 is the effective g*s that enters the baryon-to-photon ratio η_B. Both 4/11 and 43/11 are derived from entropy conservation, not assumed.

What the theorem does not claim is just as important. It does not prove that the universe actually underwent this history. The particle content (two photon polarizations, four electron-positron states, six neutrino states), instantaneous decoupling, and the two conservation assumptions remain model choices. The framework proves the arithmetic and the physics that follow once those choices are made. It also does not claim that the measured neutrino temperature has been directly confirmed to this precision; the derivation is a theoretical result, and comparison with observation is a separate empirical step.

/-- **THEOREM (provenance).** The physically derived dilution equals the
arithmetic `EntropyPerPhoton.dilutionCubed = gAfter/gBefore`: the rational
definition upstream is the value forced by entropy conservation. -/
theorem dilution_eq_dilutionCubed
    {a₁ a₂ T₁ Tγ Tν : ℝ} (ha₂ : a₂ ≠ 0) (hTγ : Tγ ≠ 0)
    (hcons : radiationEntropy 2 4 T₁ * a₁ ^ 3 = radiationEntropy 2 0 Tγ * a₂ ^ 3)
    (hfree : a₂ * Tν = a₁ * T₁) :
    (Tν / Tγ) ^ 3 = ((EntropyPerPhoton.dilutionCubed : ℚ) : ℝ) := by
  have h4 : ((EntropyPerPhoton.dilutionCubed : ℚ) : ℝ) = 4 / 11 := by
    rw [EntropyPerPhoton.dilutionCubed_eq]
    norm_num
  rw [h4]
  exact dilution_from_entropy_conservation ha₂ hTγ hcons hfree

/-- **CAPSTONE.** Entropy conservation through e± annihilation plus free
neutrino streaming force the present-day entropy density to be
`(2π²/45)·(43/11)·T_γ³`: the effective dof `g*s = 43/11` entering
`entropyPerPhoton = π⁴·g*s/(45·ζ(3))` (and hence the η_B dynamical
prefactor) is **derived** from the entropy functional, not assumed. -/
theorem gStarS_from_conservation
    {a₁ a₂ T₁ Tγ Tν : ℝ} (ha₂ : a₂ ≠ 0) (hTγ : Tγ ≠ 0)
    (hcons : radiationEntropy 2 4 T₁ * a₁ ^ 3 = radiationEntropy 2 0 Tγ * a₂ ^ 3)
    (hfree : a₂ * Tν = a₁ * T₁) :
    radiationEntropy 2 0 Tγ + radiationEntropy 0 6 Tν
      = 2 * π ^ 2 / 45 * ((EntropyPerPhoton.gStarS : ℚ) : ℝ) * Tγ ^ 3 :=
  total_entropy_eq_gStarS hTγ
    (dilution_from_entropy_conservation ha₂ hTγ hcons hfree)

What this page does not claim

The theorem does not prove that the universe's particle content or instantaneous decoupling is correct; those are model assumptions. The theorem does not claim a direct measurement of the neutrino temperature; it is a theoretical derivation. The theorem does not derive the value of the baryon-to-photon ratio itself; it only fixes the entropy degrees of freedom that enter it.

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

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