Neutron-to-Proton Freeze-Out: The Ratio That Set the Universe's Helium

What Freeze-Out Is: Equilibrium Broken by Expansion

In the first second, the universe was a plasma of photons, electrons, positrons, neutrinos, and a trace of nucleons, all at temperatures above 1 MeV (11.6 billion K). At these energies, weak interactions constantly interconverted neutrons and protons:

• n + ν_e ⇌ p + e⁻
• n + e⁺ ⇌ p + ν̄_e
• free decay: n → p + e⁻ + ν̄_e

Because these reactions ran far faster than the universe expanded, neutrons and protons stayed in thermal (chemical) equilibrium. Their relative numbers followed a Boltzmann factor set by the neutron–proton mass difference, Δm = m_n − m_p = 1.293 MeV: n/p = exp(−Δm/T). When T ≫ Δm, this ratio is essentially 1. As the universe cooled toward T ≈ Δm, protons — being lighter and cheaper to make — began to dominate.

Freeze-out is the instant the weak reaction rate per nucleon, Γ_weak, fell below the Hubble expansion rate H. Once Γ_weak < H, a given neutron could no longer find a reaction partner before the universe doubled in size, and the interconversion effectively switched off — freezing the ratio near its equilibrium value at that moment.

The Mechanism: Two Rates Crossing

Freeze-out is a competition between two temperature-dependent rates. The weak interconversion rate scales steeply with temperature because weak cross-sections grow with energy and the particle density falls with expansion:

• Weak rate: Γ_weak ∝ G_F² T⁵, where G_F is the Fermi constant.
• Expansion rate: H ∝ √(g*) · √(G_N) · T², where g* counts relativistic species and G_N is Newton's constant.

Because Γ_weak ∝ T⁵ falls faster than H ∝ T² as the universe cools, the two curves cross. Setting Γ_weak ≈ H gives the freeze-out temperature T_fr ≈ 0.7–0.8 MeV, reached at roughly t ≈ 1 s. Plugging T_fr into the Boltzmann factor:

n/p = exp(−1.293 / 0.8) ≈ exp(−1.6) ≈ 0.2 ≈ 1/5 to 1/6.

The exact value depends on a careful integration (not an instantaneous cutoff) and on g*, which depends on the number of neutrino species. This is why BBN is a sensitive probe: add extra relativistic particles, g* rises, H rises, freeze-out happens earlier and hotter, more neutrons survive, and more helium forms.

Key Numbers and a Worked Example

Start from the frozen ratio at t ≈ 1 s: n/p ≈ 1/6 (0.167). But nucleosynthesis does not begin immediately. Deuterium — the first rung on the ladder to helium — is fragile and is photo-dissociated by the enormous bath of high-energy photons. This deuterium bottleneck delays fusion until T drops to ~0.07 MeV at t ≈ 3 minutes.

During those ~180 seconds, free neutrons decay with lifetime τ_n ≈ 879 s. The surviving fraction is exp(−t/τ_n) ≈ exp(−180/879) ≈ 0.81, which lowers the ratio from 1/6 to about 1/7 (0.14).

Now the worked example. Once deuterium survives, essentially every remaining neutron is swept into ⁴He (the most tightly bound light nucleus). The neutron fraction is X_n = n/(n+p) = (1/7)/(1 + 1/7) = 1/8. Each ⁴He nucleus holds 2 neutrons and 2 protons, so the helium mass fraction is:

Y_p = 2·X_n = 2·(1/8) = 0.25.

Detailed codes give Y_p ≈ 0.247 — famously robust, and matched by observation.

How We Observe It: Reading the Fossil Helium

We cannot watch freeze-out directly, but its fingerprint survives as the primordial helium mass fraction, Y_p. Two independent lines of evidence pin it down:

• Emission-line spectroscopy of metal-poor galaxies: Astronomers measure helium and hydrogen recombination lines in the ionized gas of blue compact dwarf galaxies with very low heavy-element content, then extrapolate to zero metallicity to recover the pristine value. Modern results cluster around Y_p ≈ 0.245–0.247.
• The cosmic microwave background: The helium fraction affects the number of free electrons at recombination, subtly damping the small-scale CMB power spectrum. Planck data give a consistent Y_p, independent of the galaxy measurements.

Deuterium is the other key tracer: its abundance, measured in absorption against distant quasars, fixes the cosmic baryon density, which BBN then uses to predict Y_p. The agreement across deuterium, helium, and the CMB is one of the pillars of hot Big Bang cosmology. The pioneering calculations trace to Ralph Alpher, Robert Herman, and George Gamow in 1948, refined by Peebles and by Wagoner, Fowler, and Hoyle in the 1960s–70s.

How It Compares to Related Freeze-Outs

Neutron-to-proton freeze-out is one of several "freeze-out" events in cosmology, all governed by the same Γ vs H logic but with different physics:

• Neutrino decoupling (t ≈ 1 s, T ≈ 1 MeV): neutrinos stop interacting and stream freely, forming today's cosmic neutrino background. It happens at nearly the same instant as n/p freeze-out and directly influences it by removing a reaction channel.
• Dark matter freeze-out (the WIMP scenario): a massive particle's annihilation rate falls below H, fixing its relic abundance — the same mathematics applied to a hypothetical particle.
• Recombination / photon decoupling (t ≈ 380,000 yr): electrons combine with nuclei and the CMB is released — a chemical, not weak, freeze-out.

The crucial distinction: n/p freeze-out is a weak-interaction freeze-out whose output is a nuclear-physics quantity (the neutron budget). Unlike dark-matter freeze-out, its inputs — Δm = 1.293 MeV, G_F, τ_n, and g* — are all measured in the laboratory, which is what makes the ~25% helium prediction a genuine, parameter-free test.

Significance and Open Questions

The near-constant Y_p ≈ 0.25 is one of the most successful predictions in all of physics: a first-second calculation, using only laboratory constants, that matches galaxies and the CMB to a few percent. It rules out a cold-start universe and independently confirms the baryon density derived from the CMB.

Freeze-out is also a precision laboratory. Because more relativistic species raise H and boost helium, BBN constrains the effective number of neutrino species, N_eff ≈ 3.0, and limits exotic light particles. It even bounds time-variation of fundamental constants, since Y_p depends on Δm, G_F, and τ_n.

Two live puzzles remain. First, the neutron lifetime discrepancy: "bottle" experiments give τ_n ≈ 878 s while "beam" experiments give ~888 s — an 8–10 second gap that shifts the predicted helium and remains unresolved. Second, the primordial lithium problem: BBN predicts about three times more ⁷Li than metal-poor stars show, a persistent mismatch that hints at either stellar depletion or new physics. Neither breaks the helium story, but both keep freeze-out at the frontier.

The neutron-to-proton ratio at successive stages of the early universe, and the competing rates that drive it

Stage	Time / Temperature	n/p ratio	Dominant process
Thermal equilibrium	t < 1 s, T ≫ 1 MeV	≈ 1 (near parity)	Weak reactions fast: Γ_weak ≫ H
Freeze-out	t ≈ 1 s, T ≈ 0.8 MeV	≈ 1/6 (0.17)	Γ_weak drops below H (∝ G_F² T⁵ vs √g* T²)
Free-neutron decay era	1 s < t < ~180 s	1/6 → ~1/7 (0.14)	n → p + e⁻ + ν̄_e, τ_n ≈ 879 s
Deuterium bottleneck breaks	t ≈ 3 min, T ≈ 0.07 MeV	≈ 1/7 (0.14)	D survives; ²H, ³He, ⁴He assemble
After ⁴He locks up neutrons	t ≈ 3–20 min	n essentially all in ⁴He	Y_p = 2(n/p)/(1 + n/p) ≈ 0.25