Cosmology Neutrino Dilution Total Entropy Today

What the theorem establishes

In physical cosmology, the entropy density of the universe today is a number that encodes how much disorder, or information, is packed into each cubic centimeter of space. The standard calculation splits this into two parts: the entropy carried by photons, the particles of light, and the entropy carried by neutrinos, the nearly massless particles that stream freely through the cosmos. The Recognition Science declaration total_entropy_today establishes a precise formula for the sum of these two contributions, expressed in terms of the photon temperature T_γ and a single numerical factor.

The theorem proves that the present-day entropy density equals (2π²/45) × (43/11) × T_γ³. The number 43/11 is the effective number of entropy-carrying degrees of freedom after neutrinos have decoupled and cooled. This is not a free parameter chosen to fit data; it is forced by two physical assumptions. First, the comoving entropy of the photon-electron plasma is conserved through electron-positron annihilation. Second, neutrinos decouple before that annihilation and then redshift freely, meaning their temperature falls as the universe expands. Under these two conditions, the theorem derives the neutrino dilution factor (T_ν/T_γ)³ = 4/11, and from that the total entropy formula follows.

The power of the result is that the statistical mechanics is not assumed. The 7/8 fermion weight, the 4/3 boson law, and the 2π²/45 coefficient all come from derived integrals of the entropy functional, not from hand-picked inputs. The theorem also shows that the 43/11 factor matches the value the framework's earlier arithmetic definitions used, so those definitions are now grounded in a physical derivation rather than standing as bare rational numbers.

In Recognition Science, this is a labeled turn within a larger program. The framework's machine-checked library of formal theorems proves the statement from the two named model hypotheses. What the theorem does not claim is just as important. It does not derive the particle content of the universe: the 2 photon polarizations, 4 electron-positron degrees of freedom, and 6 neutrino degrees of freedom are inputs. It does not prove that decoupling is instantaneous, nor that entropy is conserved in the real universe; those are physical assumptions supplied by the standard cosmological model. The theorem's contribution is to show that, given those assumptions, the numerical factor 43/11 is a consequence, not a choice.

/-- **THEOREM (present-day entropy).** Photons at `T_γ` plus 6 fermionic
neutrino dof at `T_ν` with `(T_ν/T_γ)³ = 4/11` carry total entropy
`(2π²/45)·(43/11)·T_γ³`. -/
theorem total_entropy_today
    {Tγ Tν : ℝ} (hTγ : Tγ ≠ 0) (hdil : (Tν / Tγ) ^ 3 = 4 / 11) :
    radiationEntropy 2 0 Tγ + radiationEntropy 0 6 Tν
      = 2 * π ^ 2 / 45 * (43 / 11) * Tγ ^ 3 := by
  rw [radiationEntropy_eq, radiationEntropy_eq]
  have hTν3 : Tν ^ 3 = 4 / 11 * Tγ ^ 3 := by
    rw [div_pow, div_eq_iff (pow_ne_zero 3 hTγ)] at hdil
    linarith
  rw [hTν3]
  ring

/-- **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

/-- **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

/-- **THEOREM (provenance).** The present-day total equals
`(2π²/45)·gStarS·T_γ³` with `EntropyPerPhoton.gStarS`: the `43/11` upstream
is the value forced by the diluted neutrino sector. -/
theorem total_entropy_eq_gStarS
    {Tγ Tν : ℝ} (hTγ : Tγ ≠ 0) (hdil : (Tν / Tγ) ^ 3 = 4 / 11) :
    radiationEntropy 2 0 Tγ + radiationEntropy 0 6 Tν
      = 2 * π ^ 2 / 45 * ((EntropyPerPhoton.gStarS : ℚ) : ℝ) * Tγ ^ 3 := by
  have h : ((EntropyPerPhoton.gStarS : ℚ) : ℝ) = 43 / 11 := by
    rw [EntropyPerPhoton.gStarS_eq]
    norm_num
  rw [h]
  exact total_entropy_today hTγ hdil

What this page does not claim

The theorem does not derive the particle content of the universe. The theorem does not prove that decoupling is instantaneous or that entropy is conserved in the actual universe.

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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