Cosmology Neutrino Dilution

Neutrino dilution

In the standard hot big bang, the early universe was a plasma of photons, electrons, positrons, and neutrinos all at nearly the same temperature. When the universe cooled enough, electrons and positrons annihilated into photons, dumping their energy into the photon bath. The neutrinos, having already decoupled and stopped interacting, did not share that energy. The result is that today the cosmic neutrino background is cooler than the cosmic microwave background. Cosmology neutrino dilution is the name for this temperature gap, quantified by the ratio (T_ν/T_γ)³ = 4/11, where T_ν is the neutrino temperature and T_γ is the photon temperature.

The standard calculation uses conservation of entropy. Before annihilation, the coupled plasma of photons and electron-positron pairs had an effective number of entropy degrees of freedom g_before = 11/2. After annihilation, only the photons remained, with g_after = 2. Since entropy in a comoving volume is conserved, the ratio of these numbers cubed gives the dilution factor: (T_ν/T_γ)³ = g_after/g_before = 2/(11/2) = 4/11. This is textbook cosmology, found in Kolb and Turner's The Early Universe.

In Recognition Science, the framework's machine-checked library of formal theorems establishes this same result from first principles, without assuming the 4/11 or the 7/8 fermion entropy weight. The library derives the entropy density of a relativistic plasma from a fundamental entropy functional, proving the integrals that give the bosonic and fermionic contributions. From those derived integrals, a theorem shows the plasma entropy density collapses to the standard form (2π²/45)·(g_B + (7/8)g_F)·T³.

Within the framework, the dilution theorem is stated cleanly. If comoving entropy of the coupled photon-electron-positron sector is conserved through annihilation, and if decoupled neutrinos redshift freely from the shared pre-annihilation temperature, then the theorem forces (T_ν/T_γ)³ = 4/11. A second theorem then derives the present-day total entropy density as (2π²/45)·(43/11)·T_γ³, identifying the effective degrees of freedom g*s = 43/11. These are theorems in the library, proved with no gaps, over two stated model hypotheses: adiabatic expansion and free streaming.

What this means in plain language is that the framework does not take the standard cosmology numbers as inputs. The 4/11 ratio and the 43/11 effective degrees of freedom are outputs of a formal derivation from entropy conservation, with the statistical mechanics (the 7/8 weight, the 4/3 law, the 2π²/45 coefficient) all proved rather than assumed. The particle content and the two physical hypotheses remain model choices, but given those, the dilution is forced.

/-- **THEOREM (neutrino dilution from entropy conservation).**
If comoving entropy of the coupled photon–e± sector is conserved through
annihilation (`radiationEntropy 2 4 T₁ · a₁³ = radiationEntropy 2 0 T_γ · a₂³`)
and decoupled neutrinos redshift freely from the shared pre-annihilation
temperature (`a₂·T_ν = a₁·T₁`), then

  `(T_ν/T_γ)³ = 4/11`.

The dof drop `11/2 → 2` is not an input: it is produced by the derived
entropy-functional integrals inside `radiationEntropy_eq`. -/
theorem dilution_from_entropy_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) :
    (Tν / Tγ) ^ 3 = 4 / 11 := by
  rw [radiationEntropy_eq, radiationEntropy_eq] at hcons
  have hC : (2 * π ^ 2 / 45 : ℝ) ≠ 0 := by positivity
  have hkey : (11 / 2 : ℝ) * (a₁ * T₁) ^ 3 = 2 * (a₂ * Tγ) ^ 3 := by
    have h : (2 * π ^ 2 / 45 : ℝ) * ((11 / 2) * (a₁ * T₁) ^ 3)
        = (2 * π ^ 2 / 45) * (2 * (a₂ * Tγ) ^ 3) := by
      linear_combination hcons
    exact mul_left_cancel₀ hC h
  rw [← hfree] at hkey
  have ha₂3 : (a₂ : ℝ) ^ 3 ≠ 0 := pow_ne_zero 3 ha₂
  have hTT : (11 / 2 : ℝ) * Tν ^ 3 = 2 * Tγ ^ 3 := by
    have h : a₂ ^ 3 * ((11 / 2 : ℝ) * Tν ^ 3) = a₂ ^ 3 * (2 * Tγ ^ 3) := by
      linear_combination hkey
    exact mul_left_cancel₀ ha₂3 h
  rw [div_pow, div_eq_iff (pow_ne_zero 3 hTγ)]
  linarith

/-- **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)

/-- **THEOREM (structural form).** The plasma entropy density collapses to
`(2π²/45)·(gB + (7/8)·gF)·T³`, with the `2π²/45` coefficient and the `7/8`
entropy weight both coming from the derived integrals `∫σ_B = 4π⁴/45`,
`∫σ_F = 7π⁴/90` — neither is assumed. -/
theorem radiationEntropy_eq (gB gF T : ℝ) :
    radiationEntropy gB gF T = 2 * π ^ 2 / 45 * (gB + 7 / 8 * gF) * T ^ 3 := by
  unfold radiationEntropy
  rw [RadiationEntropyRelation.bose_entropy_integral_value,
    RadiationEntropyRelation.fermi_entropy_integral_value]
  have hpi : (π : ℝ) ≠ 0 := Real.pi_ne_zero
  field_simp
  ring

What this page does not claim

This answer does not claim the framework derives the particle content (2 photon polarizations, 4 electron-positron dof, 6 neutrino dof) from first principles. This answer does not claim the framework proves the physical hypotheses of adiabatic expansion and free streaming; they are model assumptions.

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)

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: