Cosmology Gstar Thresholds G Star Branch Gap High

The neutrino branch gap

In the standard model of cosmology, the number g_star counts how many relativistic particle species fill the hot early universe at a given temperature. Each species contributes its internal degrees of freedom, with fermions weighted by 7/8. At high temperature, above 200 GeV, the standard model's full content gives g_star = 106.75. The Recognition Science library implements this as a step function, g_star(T), that switches species on and off as the temperature crosses their mass thresholds.

The declaration g_star_branch_gap_high is a machine-checked theorem about two versions of this function. The baseline version assumes neutrinos are Majorana particles, with only two helicity states, contributing 6 degrees of freedom. The alternative branch treats neutrinos as Dirac particles, with right-handed states thermalized, contributing 12 degrees of freedom. The theorem proves that at 200 GeV the difference between these two branches is exactly 21/4, or 5.25. In plain terms: adopting the Dirac convention adds exactly 5.25 to the high-temperature particle count, because the extra 6 neutrino degrees of freedom enter with the fermion weight 7/8.

This gap is not a physical prediction. The library explicitly imports the neutrino convention as an input, and the theorem merely computes the arithmetic consequence of that choice. The step function itself is a leading approximation: it omits Boltzmann-suppressed tails near thresholds, the lattice QCD equation of state through the crossover, and the neutrino-decoupling reheating factor. The valid domain is T above about 1 MeV, where the approximation is standard.

What the theorem does establish is a clean, exact relationship within the model. It confirms that the high-temperature evaluation of the full step function matches the fixed 106.75 value used elsewhere in the library, and it quantifies the branch gap with rational arithmetic. A reader who wants to know how sensitive early-universe calculations are to the neutrino convention now has a precise number: 5.25, machine-checked, with no approximation error.

/-- The two neutrino conventions agree everywhere except through the
    neutrino term: the branch gap at high T is (7/8)·6 = 21/4 = 5.25. -/
theorem g_star_branch_gap_high :
    g_starWith neutrinos_dirac 200 - g_star 200 = 21/4 := by native_decide

/-- The fixed 106.75 used across the cosmology modules is the high-T
    evaluation of g_star(T): the old constant is now a function value,
    not a free-standing number. -/
theorem g_star_high_matches_derived :
    ((g_star 200 : ℚ) : ℝ) = StandardModel.RelativisticDOF.g_star_derived := by
  rw [g_star_high, StandardModel.RelativisticDOF.g_star_derived_eq]
  norm_num

/-- **g_star(T)**: the temperature-dependent relativistic degree count,
    minimal-SM neutrino convention. Instantaneous-threshold step model;
    valid for T ≳ 1 MeV. -/
def g_star (T : ℚ) : ℚ := g_starWith neutrinos T

What this page does not claim

The theorem does not predict which neutrino convention is physically correct. The step function does not model the exact crossover at the QCD confinement temperature. The library does not derive the neutrino mass thresholds from first principles.

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

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