Cosmology Number Density Integral Number Density Coeff Provenance

The provenance theorem

The number density of photons in the early universe is a standard quantity in cosmology. In natural units, it is written n_γ = (g_γ/(2π²)) T³ ∫ t²/(e^t−1) dt, where g_γ = 2 counts the two photon polarization states. The integral over the Bose-Einstein distribution has a known exact value: ∫ t²/(e^t−1) dt = 2ζ(3), where ζ(3) is Apéry's constant, approximately 1.2020569. The theorem number_density_coeff_provenance establishes that this whole coefficient, including the factor g_γ, equals exactly 2ζ(3)/π². This is not a numerical approximation; it is an identity proved in the framework's machine-checked library of formal theorems.

The proof follows a two-step Mellin transform argument. First, the series layer splits ζ(3) = Σ 1/n³ into even and odd parts, giving the Dirichlet eta function η(3) = (3/4)ζ(3). Second, the integral layer uses the Mellin transform to convert the geometric expansion of the kernel 1/(e^t−1) into a Dirichlet series at s = 3, with Γ(3) = 2. The theorem fermi_div_bose_number_integral then shows the fermion-to-boson number density ratio is exactly 3/4, the companion of the 7/8 entropy weight. This is the factor that dilutes fermionic number densities, for example the neutrino-to-photon ratio per species before dilution.

The provenance theorem is the last analytic ingredient in the entropy-per-photon formula. The capstone theorem entropyPerPhoton_from_integrals rewrites the entire ratio as a quotient of two derived thermodynamic integrals. After this step, the only model content left in the entropy-per-photon chain is the particle census (g_γ = 2, g_e = 4, g_ν = 6) and the statistical-mechanics identifications (phase-space measure, s = (4/3)ρ/T). Every analytic constant in the chain is now a theorem, axiom-clean.

What the theorem does not claim is equally precise. It does not derive the particle census or the statistical-mechanics identifications; those remain model choices. It does not provide a closed form for ζ(3), which is known to have none. It does not assert that the physical universe's photon density equals this value; that requires the model assumptions to hold. The theorem is a statement about the mathematics of the integrals, not about the cosmos itself.

/-- **THEOREM (photon number-density coefficient provenance).** The `2ζ(3)/π²`
coefficient of `n_γ = (2ζ(3)/π²)·T³` is `g_γ·(∫t²/(eᵗ−1))/(2π²)` with
`g_γ = 2`: the Bose number integral over the phase-space normalization. -/
theorem number_density_coeff_provenance :
    ((EntropyPerPhoton.gPhoton : ℚ) : ℝ)
        * (∫ t in Ioi (0 : ℝ), t ^ 2 / (Real.exp t - 1)) / (2 * π ^ 2)
      = 2 * zeta3 / π ^ 2 := by
  rw [bose_number_integral_value]
  unfold EntropyPerPhoton.gPhoton
  have hpi : (π : ℝ) ≠ 0 := Real.pi_ne_zero
  push_cast
  field_simp

/-- **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

/-- **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

What this page does not claim

The theorem does not derive the particle census or the statistical-mechanics identifications. The theorem does not provide a closed form for ζ(3), which is known to have none. The theorem does not assert that the physical universe's photon density equals this value.

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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