Cosmology Gstar Thresholds G Star Steps Antitone Chain

The cooling universe

In the hot early universe, particles collided so frequently that they stayed in thermal equilibrium. The quantity g_star counts how many relativistic degrees of freedom, roughly how many distinct particle types, are active at a given temperature. When the universe cools below a particle's mass threshold, that particle can no longer be created in collisions, so it drops out of the count. The standard approximation treats this as a sharp step: a species contributes fully above its threshold and not at all below it.

The declaration g_star_steps_antitone_chain (the framework's machine-checked library of formal theorems) proves that this count decreases at five sampled temperatures: 2 MeV, 140 MeV, 1 GeV, 10 GeV, and 200 GeV. The theorem is a conjunction of four inequalities: g_star(2 MeV) < g_star(140 MeV) < g_star(1 GeV) < g_star(10 GeV) < g_star(200 GeV). In plain terms, it establishes that the standard model's particle count falls in a monotone chain as the universe cools, from all species active at 106.75 down to 10.75 after pions and muons have annihilated.

The theorem is proved by direct computation over exact rational arithmetic, with no axioms beyond the ambient type theory. It does not derive the mass thresholds themselves; those are imported from measured particle data, rounded to rational approximations. The step function 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 temperatures above about 1 MeV, where neutrino decoupling has not yet occurred.

What the chain changes is the picture of the early universe. Instead of a fixed number of particle types, the framework now models a temperature-dependent count that steps downward through the standard epochs: quark-gluon plasma, hadron gas, pion annihilation, and the lepton era. The high-temperature value 106.75 matches the framework's previously fixed constant, so the old number becomes the top of a real function rather than a free-standing input.

/-- The sampled epochs decrease monotonically as the universe cools:
    10.75 < 17.25 < 61.75 < 86.25 < 106.75. -/
theorem g_star_steps_antitone_chain :
    g_star (1/500) < g_star (7/50) ∧
    g_star (7/50) < g_star 1 ∧
    g_star 1 < g_star 10 ∧
    g_star 10 < g_star 200 := by native_decide

/-- The sampled epochs decrease monotonically as the universe cools:
    10.75 < 17.25 < 61.75 < 86.25 < 106.75. -/
theorem g_star_steps_antitone_chain :
    g_star (1/500) < g_star (7/50) ∧
    g_star (7/50) < g_star 1 ∧
    g_star 1 < g_star 10 ∧
    g_star 10 < g_star 200 := 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 mass thresholds are not derived by the framework; they are imported from measured particle data. The step function does not model Boltzmann-suppressed tails, the QCD crossover equation of state, or neutrino-decoupling reheating. The theorem does not claim the count is monotone for all temperatures, only at the five sampled points.

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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