Cosmology Gstar Thresholds G Star Dirac High

The Dirac neutrino count

In the hot early universe, the expansion rate depends on how many particle species are relativistic and thus contribute energy density. Cosmologists package this into a single number, g_star, the effective number of relativistic degrees of freedom. The standard model with three neutrino generations gives g_star = 106.75 at temperatures above about 200 GeV, when all known particles are relativistic. That value assumes neutrinos are Majorana particles, meaning each neutrino is its own antiparticle, contributing 6 degrees of freedom.

A different convention treats neutrinos as Dirac particles, with distinct neutrinos and antineutrinos, and assumes right-handed neutrino states are also thermalized. This doubles the neutrino contribution from 6 to 12 degrees of freedom. The Recognition Science library's declaration g_star_dirac_high proves, by direct computation in its machine-checked library of formal theorems, that under this Dirac convention the high-temperature value becomes exactly 112, not 106.75. The difference is 21/4, or 5.25, which is precisely the extra 6 degrees of freedom weighted by the fermion factor 7/8.

The calculation is a piece of bookkeeping, not a physical discovery. It takes the standard model particle content as given, with masses imported from measured values, and applies the standard threshold-decoupling step function: each species contributes fully while the temperature exceeds its mass and drops out below. The declaration does not claim that neutrinos are Dirac particles, nor that right-handed states are thermalized. It only states what the count would be if those choices were made, leaving the physics of the actual neutrino sector to experiment.

The practical value of the result is that it makes the neutrino convention an explicit, checkable input rather than a hidden assumption. The same library proves that the minimal Majorana convention reproduces the textbook 106.75, so the two branches bracket the range of possible early-universe expansion rates. This matters for precision cosmology, where the effective number of relativistic species is constrained by observations of the cosmic microwave background and primordial nucleosynthesis.

/-- With thermalized right-handed Dirac neutrinos, the high-T count is
    112, not 106.75 (matches RelativisticDOF.g_star_dirac_eq). The
    neutrino convention is a real input that moves the answer. -/
theorem g_star_dirac_high : g_starWith neutrinos_dirac 200 = 112 := by
  native_decide

/-- 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

What this page does not claim

Neutrinos are Dirac particles, or that right-handed states are thermalized, as a matter of physical fact. The step-function approximation captures all finite-temperature effects, such as Boltzmann suppression near thresholds. The mass thresholds are derived from the framework's own phi-ladder predictions rather than imported from measured values.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: