Cosmology Number Density Integral Entropy Per Photon From Integrals

The entropy per photon as a ratio of integrals

In the standard hot Big Bang model, the entropy per photon is a fixed number that sets the balance between radiation and matter in the early universe. Cosmologists usually write it as a product of a particle count and a numerical factor involving π and the Apéry constant ζ(3). The Recognition Science declaration entropyPerPhoton_from_integrals proves that this number can be written instead as a ratio of two thermodynamic integrals: one for the energy density of a boson gas, the other for its number density. That is, the entropy per photon is not an isolated formula but a consequence of two deeper integrals over the distribution of particle energies.

The two integrals are standard in statistical mechanics. For a gas of bosons such as photons, the number density involves the integral ∫₀^∞ t²/(eᵗ−1) dt, which the framework's library proves equals 2ζ(3). For fermions such as neutrinos, the corresponding integral with eᵗ+1 in the denominator equals (3/2)ζ(3). Their ratio is exactly 3/4, the factor that dilutes fermionic number densities relative to bosonic ones. These are not numerical approximations; they are exact identities proved in the machine-checked library of formal theorems, with the same status as a proven lemma in a textbook.

The capstone theorem, entropyPerPhoton_from_integrals, assembles these pieces. It shows that the entropy per photon equals (4/3)·(∫ t³/(eᵗ−1))/(2π²) times the effective degrees of freedom, divided by the photon number density coefficient. After this theorem, every analytic constant in the entropy-per-photon chain is a theorem; the only remaining choices are the particle census (two photon polarizations, four electron states, six neutrino states) and the statistical identifications that define entropy density and phase space. The proof is axiom-clean, meaning it relies only on the standard axioms of the underlying logic, not on any framework-specific assumptions.

What the declaration does not claim is equally important. It does not derive the particle census; those numbers are a model choice, not a theorem. It does not prove the statistical-mechanics identifications, such as the relation s = (4/3)ρ/T; those are also model content. And it does not claim that the entropy per photon has a closed form in terms of elementary constants; the Apéry constant ζ(3) appears as a defined sum, with no closed form and none needed. The theorem's power is that it reduces the entire entropy-per-photon formula to two integrals and a handful of explicit model choices, making the analytic content fully rigorous and the remaining assumptions transparent.

/-- **THEOREM (entropy per photon from the thermodynamic integrals).**
`entropyPerPhoton` is exactly the ratio built from the two derived integrals:
numerator = entropy-density coefficient `(4/3)·(∫t³/(eᵗ−1))/(2π²)` times
`g*s`; denominator = photon number-density coefficient
`g_γ·(∫t²/(eᵗ−1))/(2π²)`. Every analytic constant in the entropy-per-photon
chain is now THEOREM; the remaining MODEL content is the particle census and
the statistical-mechanics identifications. -/
theorem entropyPerPhoton_from_integrals :
    EntropyPerPhoton.entropyPerPhoton
      = (4 / 3 * ((∫ t in Ioi (0 : ℝ), t ^ 3 / (Real.exp t - 1)) / (2 * π ^ 2))
            * ((EntropyPerPhoton.gStarS : ℚ) : ℝ))
        / (((EntropyPerPhoton.gPhoton : ℚ) : ℝ)
            * (∫ t in Ioi (0 : ℝ), t ^ 2 / (Real.exp t - 1)) / (2 * π ^ 2)) := by
  rw [FermionWeightIntegral.bose_integral_value, bose_number_integral_value,
    EntropyPerPhoton.gStarS_eq]
  unfold EntropyPerPhoton.entropyPerPhoton EntropyPerPhoton.gPhoton
  have hpi : (π : ℝ) ≠ 0 := Real.pi_ne_zero
  have hz : zeta3 ≠ 0 := zeta3_pos.ne'
  push_cast
  field_simp
  ring

/-- **THEOREM (Bose number integral).** `∫_{0}^{∞} t²/(eᵗ−1) dt = 2·ζ(3)`.
This is the analytic content of the photon number density
`n_γ = (2ζ(3)/π²)·T³`. -/
theorem bose_number_integral_value :
    (∫ t in Ioi (0 : ℝ), t ^ 2 / (Real.exp t - 1)) = 2 * zeta3 := by
  have h := mellin_bose3_value
  rw [mellin_bose3_eq_integral] at h
  exact Complex.ofReal_inj.mp h

/-- **THEOREM (Fermi number integral).** `∫_{0}^{∞} t²/(eᵗ+1) dt = (3/2)·ζ(3)`. -/
theorem fermi_number_integral_value :
    (∫ t in Ioi (0 : ℝ), t ^ 2 / (Real.exp t + 1)) = 3 / 2 * zeta3 := by
  have h := mellin_fermi3_value
  rw [mellin_fermi3_eq_integral] at h
  exact Complex.ofReal_inj.mp h

/-- **THEOREM (3/4 number-density fermion weight).** The Fermi–Dirac number
integral is exactly 3/4 of the Bose–Einstein one: `η(3)/ζ(3) = 1 − 2⁻² = 3/4`
(the companion of the 7/8 entropy weight). -/
theorem fermi_div_bose_number_integral :
    (∫ t in Ioi (0 : ℝ), t ^ 2 / (Real.exp t + 1))
      / (∫ t in Ioi (0 : ℝ), t ^ 2 / (Real.exp t - 1)) = 3 / 4 := by
  rw [bose_number_integral_value, fermi_number_integral_value]
  have hz : zeta3 ≠ 0 := zeta3_pos.ne'
  field_simp
  ring

What this page does not claim

The particle census (g_photon = 2, g_e = 4, g_nu = 6) is not derived by this theorem; it is a model choice. The statistical-mechanics identifications, including the phase-space measure and the relation s = (4/3)ρ/T, are not proved here. The theorem does not provide a closed form for ζ(3); it treats the Apéry constant as a defined sum.

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

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