rs_eta
The declaration supplies the Recognition Science baryon-to-photon ratio as the constant 6.1 times 10 to the minus 10, obtained from RS baryogenesis. Cosmologists modeling recombination or the cosmic microwave background would cite this value when inserting RS-derived inputs into the Saha equation. The entry is a bare numerical definition with no lemmas or reduction steps.
claimThe baryon-to-photon ratio equals $6.1 times 10^{-10}$.
background
The CMB Temperature module derives the present-day temperature from the recombination epoch using the Saha equation with the RS baryon-to-photon ratio. Recombination occurs near 3000 K when the ionization fraction reaches 50 percent, and the module states that the CMB temperature follows from T zero equals T star over one plus z star with z star near 1100. Upstream results include the dimensionless bridge ratio K defined as the square root of phi from the Constants module, which supports balance conditions in the framework.
proof idea
This is a direct definition that assigns the numerical value 6.1e-10 without invoking any lemmas or reduction steps.
why it matters in Recognition Science
The definition provides the RS-derived baryon-to-photon ratio required for the recombination temperature calculation that yields the CMB temperature of 2.725 K. It is referenced across the diagonal and off-diagonal components in the Metric Unification module. This constant closes the input from RS baryogenesis into the T0-T8 chain for deriving spatial dimensions and constants.
scope and limits
- Does not derive the numerical value from the forcing chain axioms.
- Does not specify the physical context or units of the ratio.
- Does not link to the J-cost or defect distance functions.
- Does not address variations in the ratio across epochs.
formal statement (Lean)
34def rs_eta : ℝ := 6.1e-10
proof body
Definition body.
35
36/-! ## Recombination Temperature -/
37
38/-- **RECOMBINATION TEMPERATURE** from Saha equation.
39 At x_e = 0.5 (50% ionization), the Saha equation gives:
40 k_B T* ≈ E_ion / ln(η⁻¹ × factor) ≈ 0.3 eV ≈ 3500 K
41
42 More precisely, T* ≈ 3000 K (accounting for detailed balance). -/