Primordial Helium Mass Fraction

What this calculator estimates

Big Bang nucleosynthesis (BBN) occurred during the first few minutes of cosmic history, when the expanding universe cooled enough for light nuclei to form. Two headline observables from this epoch are the primordial helium-4 mass fraction (Y_p) and the primordial deuterium abundance, commonly quoted as the number ratio D/H. This calculator provides a fast, classroom-friendly estimate of these quantities from three inputs: the baryon-to-photon ratio in convenient units (η₁₀), the neutron mean lifetime (τ_n), and any additional relativistic energy density beyond the Standard Model expressed as an effective neutrino contribution (ΔN_ν).

These outputs are approximations based on fitting relations to full numerical BBN calculations. They are useful for intuition and quick parameter sweeps (e.g., “what happens to Y_p if I increase ΔN_ν?”), but they are not a replacement for a modern BBN code when you need high-precision predictions and uncertainty propagation.

Key ideas behind Y_p

Helium-4 is tightly bound, so once nuclear reactions turn on efficiently (after the “deuterium bottleneck” breaks), nearly all surviving neutrons end up in ⁴He. A simple way to see why Y_p is relatively insensitive to many details is to relate it to the neutron-to-proton ratio at the onset of nucleosynthesis. Let

x = n/p

Assuming most neutrons go into ⁴He, the helium mass fraction is approximately

The ratio x is set by weak interaction freeze-out (when n↔p conversion becomes slower than expansion) and by neutron decay between freeze-out and the time nuclei begin to assemble efficiently. That’s why Y_p responds to the neutron lifetime τ_n and to any change in the expansion rate (e.g., additional relativistic species), while the baryon density mostly affects when nuclear reactions proceed and therefore has a weaker, but nonzero, impact on Y_p.

Fitting formula used for helium-4

A commonly used linearized fit around standard cosmological parameters expresses Y_p as a baseline plus small corrections from η₁₀, τ_n, and the expansion-rate factor S (which summarizes extra relativistic energy density):

Y_p = 0.2485 + 0.0016(η₁₀ − 6) + 0.0002(τ_n − 880) + 0.013(S − 1)

with

S = √(1 + 7ΔN_ν/43)

Here, η₁₀ is the baryon-to-photon ratio scaled as η₁₀ = 10¹⁰η, τ_n is the neutron mean lifetime in seconds, and ΔN_ν measures extra radiation energy density in “neutrino units” (0 corresponds to the Standard Model reference point for this simplified parameterization). Increasing ΔN_ν increases the expansion rate, generally leaving less time for neutrons to decay before nucleosynthesis, and therefore increases Y_p.

How deuterium (D/H) behaves

Deuterium is more sensitive to the baryon density than helium-4. At higher baryon density, nuclear reactions proceed more efficiently and deuterium is burned into helium more completely, so D/H decreases as η₁₀ increases. Extra radiation (higher S) can leave slightly more deuterium by speeding up expansion, but the dominant dependence is on η₁₀. Many pedagogical calculators use a power-law fit of the form D/H ∝ η₁₀^√ (roughly a negative power) with small corrections from S and τ_n.

Interpreting the results

• Y_p (helium-4 mass fraction) is a mass fraction (between 0 and 1). Typical benchmark values in standard cosmology are around 0.24–0.25. A change of 0.001 in Y_p is already meaningful in precision contexts.
• D/H is usually reported as a number ratio, often near a few × 10⁻⁵ in standard cosmology. If your result is far outside this ballpark, double-check units and whether your chosen parameters are within the validity range below.
• Sensitivity guide: changing η₁₀ moves D/H strongly and Y_p weakly; changing ΔN_ν (via S) raises Y_p noticeably; changing τ_n shifts Y_p modestly.

Worked example

Suppose you choose η₁₀ = 6.1, τ_n = 880 s, and ΔN_ν = 0.

1. Compute S: S = √(1 + 7×0/43) = 1.
2. Plug into the helium fit:
   • Baseline: 0.2485
   • Baryon term: 0.0016(6.1−6) = 0.00016
   • Neutron lifetime term: 0.0002(880−880) = 0
   • Radiation term: 0.013(1−1) = 0

So Y_p ≈ 0.2485 + 0.00016 = 0.24866. Interpreted as a percentage, that is about 24.866% of the baryonic mass in helium-4.

Parameter effects at a glance

Assumptions, validity range, and limitations

• Approximate fits: The calculator uses simplified fitting relations intended to reproduce the behavior of full numerical BBN calculations near standard parameters. Away from the calibration region, errors can grow and nonlinear effects can matter.
• Parameter range: Treat results as most reliable for η₁₀ in the rough neighborhood of a few to ~10, τ_n near current experimental values (~870–890 s), and small-to-moderate ΔN_ν. Extremely large ΔN_ν can invalidate the linearized helium correction and the simple S mapping.
• Model dependence: Real BBN predictions depend on nuclear reaction rates (with uncertainties), neutrino physics, and details of the thermal history. This calculator does not propagate those uncertainties.
• Notation: ΔN_ν here is a convenience parameter for extra relativistic energy density; different papers distinguish between N_eff, ΔN_eff, and related definitions. Ensure consistent definitions when comparing to a specific dataset or publication.
• Not a measurement tool: Observational inferences of primordial Y_p and D/H include astrophysical systematics (e.g., chemical evolution, ionization corrections). This calculator outputs theoretical estimates only.

References for context (non-exhaustive)

For authoritative background, see standard BBN reviews and textbooks (e.g., Steigman’s BBN reviews; particle data compilations summarizing primordial abundances; and modern numerical BBN code papers). If you need publication-grade predictions, use a current BBN code and up-to-date nuclear rates.