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Primordix

Big Bang Nucleosynthesis / Primordial Helium Physics

\[Y_p \simeq \frac{2\,(n/p)}{1 + n/p}\]
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Primordix is a Pixelated Physics law character, one of our relativity and cosmos designs. In the first minutes after the Big Bang, about a quarter of ordinary mass became helium-4 before many free neutrons decayed. This is a 3" × 3" pixel-art sticker. The physics panel below gives the equation, what each symbol means and the sources.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

In the first minutes after the Big Bang, the ratio of neutrons to protons froze out at a small fixed value. Nearly all those neutrons ended up in helium-4, which predicts the fraction of ordinary mass that came out as helium, matching observations.

Primordix is a baker with the worst kitchen in history and an ingredient that spoils.

In the first second after the Big Bang, the universe is too hot for nuclei. Protons and neutrons turn into each other freely through the weak force. As it cools, those reactions freeze out, and because a neutron outweighs a proton by 1.293 MeV, the last line on this page, neutrons end up outnumbered, about one for every six protons at freeze-out. Then the neutrons start to decay.

She can’t bake yet. Any deuterium she forms, the first step towards helium, is blasted apart by photons that outnumber baryons more than a billion to one. She has to wait until the oven cools, a few minutes in. By then decay has cut the ratio to about one neutron per seven protons. Then she bakes fast: almost every surviving neutron ends up in helium-4. The first two lines turn that into a helium share of about 0.25 by mass. Observations give 0.245 ± 0.003; standard theory with Planck’s baryon density predicts about 0.247. That is a quarter by mass; by number of nuclei, helium is only about 7.5%.

How fast her neutrons spoil is still argued. Experiments that trap ultracold neutrons in bottles give a mean life of about 878 s (Particle Data Group 2024: 878.4 ± 0.5 s). Experiments that count decays in a neutron beam give about 888 s. The gap is unresolved and remains an open question. Lithium-7 is another open problem: theory predicts about three times what old stars show.

Ralph Alpher, Hans Bethe and George Gamow proposed cooking the elements in a hot early universe in 1948; the modern calculation came later.

Three minutes. Neutrons won’t wait.

Equations

\[Y_p \simeq \frac{2\,(n/p)}{1 + n/p}\]
\[\begin{gathered} \frac{n}{p} \approx \frac{1}{7} \;\Rightarrow \\ Y_p \approx \frac{2/7}{8/7} = 0.25 \end{gathered}\]
\[\left.\frac{n}{p}\right|_{\text{freeze}} \approx e^{-Q/k_B T_f}\]
\[\begin{aligned} Q &= (m_n - m_p)\,c^2 \\ &= 1.293\ \text{MeV} \end{aligned}\]
Symbols
SymbolMeaningUnit
\(Y_p\) primordial helium-4 fraction by mass (observed 0.245 ± 0.003) \(\mathrm{1}\)
\(n/p\) neutron-to-proton ratio \(\mathrm{1}\)
\(Q\) rest-energy difference between neutron and proton \(\mathrm{MeV}\)
\(k_B\) Boltzmann constant \(\mathrm{eV/K}\)
\(T_f\) temperature when the weak reactions freeze out (thermal energy about 0.7–0.8 MeV) \(\mathrm{K}\)
\(m_n, m_p\) neutron and proton masses \(\mathrm{kg}\)
\(c\) speed of light in vacuum \(\mathrm{m/s}\)

Source Peebles, P. J. E. (1966). Primordial Helium Abundance and the Primordial Fireball. II. The Astrophysical Journal 146, 542.doi.org/10.1086/148918 (opens in a new tab)