Primordial Nucleosynthesis

Early formation of light nuclei and cosmological abundances. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Primordial nucleosynthesis describes the short stage when the hot, dense universe assembled protons and neutrons into light nuclei. It was not a factory for all elements: it mainly produced hydrogen, deuterium, helium-3, helium-4, and traces of lithium. The central question is why $Y_p$ settles near a stable value and how $\eta$ determines that value. The key is a cosmic race: expansion cools the plasma, $T$ allows or blocks reactions, deuterium must survive, and free neutrons are lost as $t$ advances.

🟢 Essential level

$\eta$ tells how much nuclear matter is available compared with the relic light bath.

If there are more baryons per photon, $n_b$ is larger for the same $n_{\gamma}$, nuclear collisions become more likely, and the chain passing through deuterium can proceed before expansion freezes it. Yet $Y_p$ does not rise like an open tap: it depends mainly on how many neutrons remain. $r_np$ is first set by thermal equilibrium and then decreases because free neutrons decay during $t$. When deuterium stops being destroyed, almost all surviving neutrons end up in helium-4. Therefore $\eta$ gently shifts the result rather than controlling it linearly, and it should be read as a lever of nuclear opportunity.

Primordial nucleosynthesis took place in a window of just a few minutes, between the first second and about twenty minutes after the Big Bang. That brief interval left a measurable fingerprint today: the $Y_p$ observed in primordial gas, around 0.25, agrees remarkably well with the standard model prediction, making it one of the strongest confirmations of hot Big Bang cosmology.

🔵 Formal level

Basic formulas

The model is built by chaining several relations with different roles. First, $T$ and Delta_m_np set $r_np$ while weak interactions maintain thermal equilibrium. Here $k_B$ converts temperature into energy, so the exponent must be dimensionless and the negative sign expresses neutron suppression.

Neutron-proton ratio in thermal equilibrium

Then, between weak freeze-out and efficient nucleosynthesis, free neutrons decay during $t$ with a scale given by $\tau_{n}$. This stage reduces the neutron supply and prepares the later helium estimate.

Free neutron survival

Real student questions

Why were heavy elements not formed? Because the window of $T$ and $t$ was short, and stable nuclei with certain mass numbers are missing, blocking a continuous ladder.

Is $\eta$ a density? No. It is a dimensionless ratio; it combines with $n_{\gamma}$ to obtain $n_b$.

Why do almost all neutrons end in helium-4? Because helium-4 is strongly bound and becomes a stable nuclear sink once the deuterium bottleneck opens.

Is $Y_p$ exact? No. The leaf formulas give physical reading and approximations; precision prediction requires nuclear networks, reaction rates, and cosmological parameters.

Final synthesis

$\eta$ determines $Y_p$ by changing baryon density and the moment when the nuclear network can act. But helium-4 is governed by a deeper piece: how many neutrons survive when deuterium opens the gate.