About a quarter of the ordinary matter in the universe is helium. Where did it come from? Stars make helium, as in the proton–proton chain, but not nearly enough. Most of it was made in the first few minutes, when the whole universe was as hot as a stellar core, a time we can reach by running the background radiation's temperature back.
Why it works
Freeze-out. At first, weak reactions turned neutrons and protons into each other freely, keeping $n/p = e^{-\Delta mc^2/kT}$ with $\Delta mc^2 = 1.293$ MeV. About one second after the Big Bang, at $kT \approx 0.8$ MeV, these reactions became too slow to keep up, and the ratio froze near $e^{-1.293/0.8} \approx 1/5$ (a fuller calculation, which follows the reactions as they slow, gives about 1/6). Decay. Free neutrons decay (mean lifetime 879 s), so the ratio drifted down. The deuterium bottleneck. Deuterium is the first step to helium, but until the temperature fell to about $10^{9}$ K, energetic photons broke it apart as fast as it formed. About 3 to 4 minutes in, with $n/p$ down to about $1/7$, deuterium survived and nearly every neutron was swept into helium-4. With 2 neutrons and 2 protons per helium nucleus, the mass fraction is $$Y = \frac{2(n/p)}{1 + n/p} = \frac{2/7}{8/7} = 0.25.$$
Key idea
Big Bang nucleosynthesis made about 25% helium by mass, plus traces of deuterium, helium-3 and lithium, in the first few minutes. The oldest gas has about 24 to 25% helium, and its deuterium matches the amount of ordinary matter measured from the microwave background. Stars have added only a few percent of helium since.
Worked examples
Worked example: Helium from a neutron count
With 1 neutron for every 7 protons, what mass fraction ends up in helium-4?
- Take 16 nucleons: 2 neutrons, 14 protons.
- One helium-4 uses the 2 neutrons and 2 protons: 4 of 16 nucleons.
Answer: $Y = 4/16 = 0.25$.
Worked example: Freeze-out
Find $n/p$ at $kT = 0.8$ MeV.
- $1.293/0.8 = 1.62$.
Answer: $n/p = e^{-1.62} = 0.20$, about 1/5.
Worked example: Could stars have made the helium?
Suppose the Milky Way has shone at $2.5 \times 10^{10}\,L_\odot$ for 13 Gyr, all from turning hydrogen into helium ($0.0071c^2$ per kilogram). How much helium could that make, compared with its $6 \times 10^{10}\,M_\odot$ of stars and gas?
- Power: $2.5 \times 10^{10} \times 3.83 \times 10^{26} = 9.6 \times 10^{36}$ W. Time: $13 \times 10^{9} \times 3.16 \times 10^{7} = 4.1 \times 10^{17}$ s. Energy: $3.9 \times 10^{54}$ J.
- Mass processed: $3.9 \times 10^{54}/(0.0071 \times 9.00 \times 10^{16}) = 6.2 \times 10^{39}$ kg $= 3.1 \times 10^{9}\,M_\odot$.
Answer: About 5% of the mass, a rough estimate but far short of the 25% observed: most helium is primordial.
Common mistake
Thinking all of the helium in the universe was made in stars. Stars make helium, but even a whole galaxy shining for its entire life converts only a few percent of its mass. And helium is about 24% of the mass even in the oldest stars and the most pristine gas, which stars have hardly touched. Check with the estimate above: compare the energy a galaxy has radiated with the energy needed to make a quarter of its mass into helium.
The amounts of deuterium and helium-3 predicted by the theory match observations well. Lithium-7 is the exception: old stars show about three times less than predicted, a puzzle that is still being studied.