Cosmology Gstar Thresholds G Star 1 Ge V

Counting the early universe

In the early universe, temperature dictates which particles exist. The quantity g_star, the effective number of relativistic degrees of freedom, is the standard bookkeeping device for this: at a given temperature, each particle species contributes a fixed count if it is light enough to be created, and drops out once the universe cools below its mass threshold. At one billion electronvolts, the count is 61.75.

The number comes from a machine-checked library of formal theorems that implements the standard threshold-decoupling step function. Each species, from photons to the top quark, carries a mass threshold and a degree count; fermions contribute seven-eighths of their spin states, bosons contribute fully. At 1 GeV, the bottom, tau, and charm quarks have already decoupled, leaving a plasma of gluons, up, down, and strange quarks, plus photons, leptons, and neutrinos. The theorem g_star_1GeV proves that the sum is exactly 247/4, or 61.75.

This is a recognition framework model, a definitional choice built from imported Standard Model content, not a derived law. The step function is the leading approximation: it ignores Boltzmann-suppressed tails near thresholds, the lattice QCD equation of state through the crossover, and the neutrino-decoupling reheating factor. The valid domain is temperatures above about 1 MeV, where neutrino decoupling matters. Mass thresholds are imported from measured PDG values, rounded to rational approximations; the framework's own phi-ladder mass predictions are not used here.

The value matters because it sets the expansion rate of the early universe. More active species mean more energy density, which means faster cooling and a different timeline for nucleosynthesis and structure formation. The framework's contribution is a clean, machine-checked implementation of the standard calculation, not a new physics prediction. It also verifies consistency: at high temperatures, the step function matches the framework's fixed 106.75 value, so the old constant is now a special case of a real function.

/-- T = 1 GeV: b, τ, c also decoupled → 247/4 = 61.75. -/
theorem g_star_1GeV : g_star 1 = 247/4 := by native_decide

/-- T = 1 GeV: b, τ, c also decoupled → 247/4 = 61.75. -/
theorem g_star_1GeV : g_star 1 = 247/4 := 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

The value 61.75 is not a derivation from the recognition framework's axioms; it is a model built from imported Standard Model content. The step function does not account for Boltzmann-suppressed tails or the finite-temperature equation of state. The framework's phi-ladder mass predictions are not used here; the thresholds are imported PDG 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)

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