---
title: "Big Bang Nucleosynthesis: Light Elements in the First Minutes Interactive Lab | Sandbox Physics"
description: "How do temperature and expansion speed decide today’s helium and deuterium? Expansion sets the clock and temperature sets the gate."
canonical_url: "https://sandboxphysics.com/labs/big-bang-light-element-network/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/big-bang-light-element-network/"
last_modified: "2026-11-14"
---

# Big Bang Nucleosynthesis: Light Elements in the First Minutes

**How do temperature and expansion speed decide today’s helium and deuterium?** Expansion sets the clock and temperature sets the gate. Neutrons stop turning into protons about one second after the big bang, then decay until the universe is cool enough, three to four minutes later, for deuterium to survive. Almost every neutron left at that moment ends up in helium. Denser baryons burn more deuterium away, and faster expansion leaves more neutrons and more helium.

Drag the cosmic clock through the first hours and watch free neutrons decay, wait, then lock into helium. Change the baryon density and the extra radiation to see the final helium, deuterium and lithium move.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The model follows eight light species through twelve reactions from 8.6 MeV to 0.86 keV. Nuclear rates are the fitted rates of Serpico et al. (2004), used inside fixed temperature windows and extended outside them by a Gamow law or a constant. Neutron-proton rates use the Born approximation normalised to a neutron lifetime of 878.4 seconds, without radiative, finite-nucleon-mass or weak-magnetism corrections. Neutrinos decouple instantly before electron-positron annihilation, so three neutrinos give an effective number of exactly three rather than 3.044, and extra radiation adds neutrino-like energy density. Chemical potentials and plasma corrections are zero. These choices put the predicted helium fraction about 0.005 below full calculations. Observed abundances are the PDG 2025 values; the model curves are separate predictions. The drawn box holds 400 nucleons to scale, so deuterium, helium-3 and lithium are too rare to appear in it.

## What the first hours leave behind

Neutrons $n$ Protons $p$ Deuterium $\mathrm{D}$ ${}^{3}\mathrm{He}+{}^{3}\mathrm{H}$ Helium-4 ${}^{4}\mathrm{He}$ ${}^{7}\mathrm{Li}+{}^{7}\mathrm{Be}$

Mass fractions through time

Model calculation

Neutrons drop at once when helium forms.

Final abundances against baryon density

Model curves; grey bands observed (PDG 2025); green band from the microwave background

Lithium-7 comes out about three times above stars.

Which nuclear rates move the result

Model; each rate moved by its one-sigma error (Serpico 2004)

$\mathrm{d}(p,\gamma){}^{3}\mathrm{He}$ sets most of the deuterium error.

## Follow the neutrons from one second to three minutes

**Background** A second after the big bang the universe is a hot soup of photons, electrons, positrons, neutrinos and a thin sprinkling of neutrons and protons. Weak reactions turn neutrons and protons into each other, and photons break apart any deuteron that forms.

**Why it matters** Almost all of today’s helium and deuterium was made in these few minutes, so their abundances record how hot the universe was, how fast it expanded and how many baryons it held.

- **Focus question**: Why does nearly a quarter of the ordinary mass end up as helium, and why is a little deuterium left over?
- **One-sentence intuition**: Helium counts the neutrons that survive until deuterium can hold together. Expansion sets that waiting time, so more radiation means more helium. Deuterium is the leftover fuel: more baryons burn it further, which is why $\mathrm{D/H}$ measures the baryon density.

### Core mathematical model

#### Neutrons and protons stop trading

$$
\left.\frac{n}{p}\right|_{\mathrm{eq}}=\exp\!\left(-\frac{Q}{k_BT}\right),\qquad Q=(m_n-m_p)c^2=1.293\,\mathrm{MeV}
$$

While weak reactions are fast, the ratio follows the temperature. Near $k_BT\approx0.7\,\mathrm{MeV}$ they become slower than the expansion. The ratio stops following the temperature at about one neutron for every five protons, then falls slowly because free neutrons decay with lifetime $\tau_n\approx879\,\mathrm{s}$.

#### Expansion rate from the radiation

$$
H^2=\frac{8\pi G}{3}\,\frac{\pi^2}{30}\,g_*\frac{(k_BT)^4}{(\hbar c)^3c^2},\qquad g_*=2+\frac{7}{8}\left[4+2\left(3+\Delta N_{\mathrm{eff}}\right)\left(\frac{T_\nu}{T}\right)^{4}\right]
$$

Before electron-positron pairs annihilate, photons, pairs and neutrinos fill the energy density. Each extra neutrino-like species $\Delta N_{\mathrm{eff}}=1$ raises the expansion rate, so weak reactions freeze earlier, fewer neutrons decay, and more helium forms. In this model the helium fraction rises by about 0.013 per extra species.

#### Why deuterium waits

$$
\frac{X_d}{X_nX_p}\propto\eta\left(\frac{k_BT}{m_Nc^2}\right)^{3/2}\exp\!\left(\frac{B_d}{k_BT}\right),\qquad B_d=2.224\,\mathrm{MeV}
$$

There are more than a billion photons for every baryon, so even when $k_BT$ is far below the deuteron binding energy, the rare energetic photons still break deuterons apart. The small factor $\eta$ delays deuterium until about $k_BT\approx0.07\,\mathrm{MeV}$, a few minutes after the big bang. Only then can the chain to helium start.

#### Helium counts the surviving neutrons

$$
Y_p\approx\frac{2\,(n/p)}{1+n/p}
$$

Nearly every neutron still free when the bottleneck opens ends up in helium-4, which holds two neutrons and two protons. With $n/p\approx1/7$ this gives $Y_p\approx0.25$. The full network in this Lab gives 0.242 for three neutrino species, which corresponds to $n/p\approx0.138$ when the bottleneck opens.

### Common difficulties

#### Cooling below the binding energy is not enough

**Typical misconception** Deuterium forms as soon as the temperature drops below its binding energy of 2.2 MeV.

**Better mental model** Photons outnumber baryons by more than a billion to one. The high-energy tail of the photon distribution keeps breaking deuterons until the temperature is about thirty times lower.

#### More baryons do not mean more deuterium

**Typical misconception** A denser universe should make more of every element.

**Better mental model** Helium rises only slightly with density, while deuterium falls steeply because denser matter burns it into helium-3 and helium-4 faster. The sparse-baryon preset leaves about six times more deuterium than our universe.

#### The lithium curve is a prediction, not a fit

**Typical misconception** The model is tuned so that every element matches observation.

**Better mental model** Only the baryon density and the radiation are inputs, and they are fixed independently by the cosmic microwave background. With them, helium and deuterium agree with observation, but lithium-7 comes out about three times higher than in old halo stars. That gap is still unexplained.

### Run the experiment

1. 01

   #### Watch one run of the first hours

   Let Our universe play from the start, then drag along the cosmic clock to about 200 seconds.

   **What to observe:** Blue neutrons fade slowly at first. Around three to four minutes they pair with protons and the box fills with gold helium nuclei in a few tens of seconds.

2. 02

   #### Change the baryon density

   Compare Sparse baryons keep deuterium with Dense baryons burn deuterium, then drag the baryon slider slowly between them.

   **What to observe:** $\mathrm{D/H}$ drops by a factor of about thirty from the first to the second, while the helium fraction only moves from 0.230 to 0.250. The observed deuterium band meets the curve just beside the green microwave-background band.

3. 03

   #### Speed up and slow down the expansion

   Compare Extra radiation, more helium with Slower expansion, less helium.

   **What to observe:** The neutron-to-proton ratio at $10^{9}\,\mathrm{K}$ moves from 0.143 to 0.165 and the helium fraction from 0.228 to 0.261. One extra species shifts helium by about four times the observed one-sigma error, so only a small change in radiation is allowed.

4. 04

   #### Find the rates that matter

   Return to Our universe and read the chart of nuclear rates below the scene.

   **What to observe:** Moving $\mathrm{d}(p,\gamma){}^{3}\mathrm{He}$ by its published error changes deuterium by about 0.7 percent, and lithium-7 responds most to that rate and to ${}^{3}\mathrm{He}(\alpha,\gamma){}^{7}\mathrm{Be}$. These errors are far too small to close the lithium gap.

Physical conclusion

Two numbers fix the light elements: how many baryons there are for each photon and how fast the universe expanded. Deuterium measures the first, helium constrains the second, and lithium-7 is the one element whose prediction still disagrees with observation.

## Related experiments and articles

The first minutes and the light elements · Reading path

### From the first second to the light elements

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How temperature and expansion speed set helium and deuterium**](https://sandboxphysics.com/stories/primordial-nucleosynthesis/temperature-expansion-and-light-elements/)
2. [Article **When neutrons and protons stop turning into each other**](https://sandboxphysics.com/stories/primordial-nucleosynthesis/when-neutrons-and-protons-stop-converting/)
3. [Article **Why deuterium waits for the universe to cool**](https://sandboxphysics.com/stories/primordial-nucleosynthesis/why-deuterium-waits/)
4. [Article **How baryon density and extra radiation move the light elements**](https://sandboxphysics.com/stories/primordial-nucleosynthesis/baryon-density-radiation-and-abundance-sensitivity/)
5. [Article **Recompute a small network from a frozen rate grid and propagate rate errors**](https://sandboxphysics.com/stories/primordial-nucleosynthesis/recompute-a-frozen-rate-network/)
6. [Article **Reading the BBN evidence on extra radiation and the lithium problem**](https://sandboxphysics.com/stories/primordial-nucleosynthesis/reading-lithium-and-extra-radiation-bounds/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)