Cosmology Gstar Thresholds G Star 2 Me V

Counting the early universe's contents

In the standard model of cosmology, the hot early universe was a plasma of particles in thermal equilibrium. Cosmologists track how many kinds of particles were active at a given temperature with a quantity called g_star, the effective number of relativistic degrees of freedom. It is a weighted headcount: each particle species contributes to the count while the temperature is high enough to keep it relativistic, and drops out once the plasma cools below its mass threshold. The function g_star(T) therefore falls in steps as the universe expands and cools, each step marking a species becoming non-relativistic and effectively disappearing from the thermal bath.

At a temperature of 2 MeV, the universe had already passed several such steps. Pions and muons, with masses near 140 MeV and 106 MeV respectively, had annihilated. The remaining relativistic species were photons, electrons, and three generations of neutrinos. The standard textbook count for this epoch is 10.75, a value that appears in cosmology textbooks as part of the standard thermal history. The Recognition Science library contains a machine-checked proof of this value: its declaration g_star_2MeV establishes that the step function g_star, evaluated at 2 MeV, equals 43/4, which is 10.75 in decimal form.

The library's g_star function is a deliberately simple model. It treats each species as either fully active or fully absent, switching abruptly at the mass threshold. This ignores the smooth Boltzmann tails near thresholds, the details of the QCD crossover, and the neutrino-decoupling reheating factor. The valid domain is temperatures at or above about 1 MeV, the regime where the approximation is standard. The particle masses themselves are imported from measured values, not derived within the framework; the function only uses their relative order against the temperature.

What the declaration does not claim is more interesting than what it proves. It does not derive the particle masses from first principles, it does not model the smooth decoupling of species, and it does not compute the full equation of state of the early universe. It is a spot check that the framework's implementation of the standard step-function approximation reproduces a known textbook value at one specific temperature. The proof itself is a direct computation, checking that the sum of active species' contributions at 2 MeV equals 43/4.

The value 10.75 matters because it feeds directly into the expansion rate of the early universe. The energy density of relativistic particles scales with g_star, and the expansion rate scales with the square root of the energy density. A higher g_star means a faster expansion and a younger universe at a given temperature. The step function, despite its simplicity, is the standard leading approximation used in nucleosynthesis calculations, and this declaration confirms the framework's implementation agrees with the conventional count at the 2 MeV epoch.

/-- T = 2 MeV (above neutrino decoupling; π, μ gone): γ, e, ν
    → 43/4 = 10.75. -/
theorem g_star_2MeV : g_star (1/500) = 43/4 := by native_decide

/-- Species relativistic and populated at temperature T (GeV), with the
    neutrino sector supplied as an explicit input (minimal vs. Dirac). -/
def activeWith (nu : Species) (T : ℚ) : List Species :=
  [photon, nu]
    ++ ew_species.filter (fun s => s.mass < T)
    ++ (if T_qcd < T then qgp_species
        else hadron_species.filter (fun s => s.mass < T))

/-- T = 2 MeV (above neutrino decoupling; π, μ gone): γ, e, ν
    → 43/4 = 10.75. -/
theorem g_star_2MeV : g_star (1/500) = 43/4 := by native_decide

/-- T = 2 MeV (above neutrino decoupling; π, μ gone): γ, e, ν
    → 43/4 = 10.75. -/
theorem g_star_2MeV : g_star (1/500) = 43/4 := by native_decide

/-- A thermal species: name, mass threshold (GeV, rational approximation;
    only its order relative to T matters), internal degrees of freedom,
    and quantum statistics. -/
structure Species where
  name : String
  mass : ℚ
  dof : ℕ
  fermion : Bool

/-- **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 declaration does not derive the particle masses from the framework's phi-ladder; it imports them as measured inputs. It does not model the smooth decoupling of species near thresholds, only the abrupt step approximation. It does not compute the full equation of state or the expansion history of the early universe.

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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