Primordial Helium and Neutrinos: What BBN Already Knew

The cosmic microwave background gets most of the attention as a probe of fundamental physics in the early universe — and rightly so, because its precision is breathtaking. But there is an older, lower-temperature probe that constrained the neutrino sector decades before CMB anisotropies were ever measured: Big Bang nucleosynthesis (BBN), the formation of the lightest nuclei during the first few minutes after the Bang.

BBN is older as a constraint and BBN is also colder. The CMB last scattered at a temperature of about 0.3 eV, hundreds of thousands of years after the Big Bang. BBN unfolded much earlier and much hotter — between about 1 MeV and 50 keV, in the universe’s first three minutes. At those temperatures, the universe was hot enough that weak reactions kept neutrons and protons in equilibrium, hot enough that neutrinos were thermal and relativistic, and hot enough that nuclear reactions could fuse protons and neutrons into deuterium and then into helium. The abundances that emerged depend sensitively on what was going on in the neutrino sector at the time, and the agreement between BBN predictions and the helium and deuterium we actually observe is one of the great triumphs of the hot Big Bang model.

This post is about how BBN encodes neutrino physics, what it tells us that the CMB does not, and why it remains an essential cross-check on the standard cosmological history.

The clock and the freeze-out

The expansion rate of the early universe, set by the Friedmann equation, is

$H^2=\frac{8\pi G}{3}\rho$

where $\rho$ is the total energy density. In the radiation era this is dominated by relativistic species — photons, electrons, positrons and neutrinos — and the density is conventionally written

${\rho }_{\mathrm{rad}}=\frac{{\pi }^2}{30}{g}_{\ast }{T}^4$

with $g_{\ast }$ the effective number of relativistic degrees of freedom. Each additional neutrino species adds $7/8\times 2=7/4$ to $g_{\ast }$ (the $7/8$ is the fermion factor and the 2 counts particle and antiparticle), so the standard three-flavor neutrino sector contributes $21/4$ on top of photons and electrons.

Around a temperature of 1 MeV the weak interactions that interconvert neutrons and protons —

$n+{\nu }_e\leftrightarrow p+e^{-},n+e^{+}\leftrightarrow p+{\bar{\nu }}_e,n\leftrightarrow p+e^{-}+{\bar{\nu }}_e$

— freeze out, because the weak rate drops below the expansion rate. From that moment on, the ratio of neutrons to protons is locked in at approximately

${\left(\frac{n}{p}\right)}_{\mathrm{freeze}-\mathrm{out}}\approx \exp \left(-\frac{\Delta m}{T_{\mathrm{f}}}\right)\approx \frac{1}{6}$

where $\Delta m=m_n-m_p\approx 1.29$ MeV is the neutron-proton mass difference and $T_{\mathrm{f}}\approx 0.7$ MeV is the freeze-out temperature.

Anything that affects the expansion rate or the weak rates around 1 MeV moves this ratio. If the universe expands faster — for instance because $g_{\ast }$ is larger thanks to extra relativistic species — freeze-out happens at a higher temperature, the exponential is less suppressed, and more neutrons survive. The single most consequential downstream effect of more neutrons is that more helium-4 is built when the nuclear reactions later kick in, because essentially every available neutron ends up bound in a helium nucleus.

The deuterium bottleneck

After the n/p ratio freezes, neutrons begin to decay with a lifetime of about 880 seconds. The universe is in a race: build neutrons into nuclei before they all decay. The reaction that has to start the chain is

$n+p\to d+\gamma$

but the deuteron is loosely bound — only 2.2 MeV — and at temperatures above about 0.07 MeV the ambient gamma rays in the photon-baryon plasma are energetic enough to immediately photodissociate it. This is the deuterium bottleneck.

The universe has to cool below the bottleneck before fusion can really get going. Once it does, the reaction chain races forward: deuterium captures another neutron or proton to make helium-3 or tritium, and these fuse into helium-4 essentially as fast as deuterium becomes available. The result is that virtually every neutron that survived until the bottleneck ends up bound in helium-4, with a tiny tail going into deuterium, helium-3, and a still tinier trace of lithium-7. The final helium abundance, $Y_p\approx 0.245$, is largely set by the n/p ratio at the time the bottleneck breaks.

What BBN sees about neutrinos

Two distinct neutrino effects are encoded in the abundance pattern.

The first, more direct effect is expansion-rate sensitivity. Any change in $g_{\ast }$ around 1 MeV shifts the freeze-out temperature and changes $Y_p$. This is conveniently parameterized as the effective neutrino number $N_{\mathrm{eff}}$, defined so that $N_{\mathrm{eff}}=3$ corresponds to the standard Standard-Model neutrino contribution. A small departure from 3 — extra sterile neutrinos, late decaying particles dumping entropy into the radiation, modifications of the neutrino sector — would push $Y_p$ up or down. Modern BBN analyses, combining precise helium-abundance measurements from low-metallicity HII regions with deuterium abundances from high-redshift quasar absorption systems, find

$N_{\mathrm{eff}}^{\mathrm{BBN}}=2.86\pm 0.15$

consistent with three flavors. This BBN bound is one of the strongest direct experimental constraints on light, sterile species and was one of the first cosmological arguments against a four-flavor neutrino sector.

The second effect, more subtle, is the electron-neutrino spectrum. The weak rates that govern n/p equilibrium involve electron neutrinos and antineutrinos directly. If their spectra are distorted — for example, by partial decoupling effects, by neutrino oscillations into sterile states before freeze-out, or by exotic interactions — the n/p ratio at freeze-out shifts even at fixed $N_{\mathrm{eff}}$. Standard calculations now include the small non-thermal distortion of the electron-neutrino spectrum from incomplete decoupling, and this is part of why the “exact” Standard-Model prediction is $N_{\mathrm{eff}}=3.044$ rather than $3.000$. Anything beyond that shifts the predicted abundances in calculable ways.

Why BBN and the CMB are complementary

The CMB constrains $N_{\mathrm{eff}}$ at the time of recombination, hundreds of thousands of years after BBN. Planck’s value, $N_{\mathrm{eff}}=2.99\pm 0.17$, is in excellent agreement with BBN. But the two probes are sensitive to different epochs and different physics, so the agreement is a non-trivial cross-check.

A scenario in which a heavy particle decays between BBN and recombination, dumping entropy into the radiation, would raise $N_{\mathrm{eff}}^{\mathrm{CMB}}$ above $N_{\mathrm{eff}}^{\mathrm{BBN}}$. A scenario in which neutrino self-interactions or coupling to dark sectors modify the free-streaming behavior would show up in the CMB acoustic peaks but not in BBN. Conversely, a scenario that changed neutron-proton equilibrium without changing $g_{\ast }$ would affect BBN but barely touch the CMB. Combining the two, with their independent systematics and physics, leaves much less room for new physics in either epoch than either probe alone.

What it constrains, what it doesn’t

BBN’s strengths are its early epoch and its sensitivity to relativistic energy density. It is essentially blind, however, to neutrino masses — which are far too small to matter at the temperatures of nucleosynthesis. The CMB and large-scale structure are where the absolute neutrino mass scale is probed, through its effect on structure growth.

BBN is also limited by observational systematics. The primordial helium abundance must be inferred from spectroscopy of metal-poor extragalactic regions, with corrections for stellar contamination and ionization. The primordial deuterium abundance comes from a small number of quasar absorption systems and depends on careful modelling of intervening clouds. The remarkable thing is how robustly the standard BBN picture has held up as these measurements have improved over decades: the predictions and the observations have stayed in agreement at the few-percent level.

The lithium-7 abundance is the one persistent disagreement. Standard BBN predicts about three times more lithium-7 than is observed in the atmospheres of metal-poor halo stars — the so-called lithium problem. Stellar destruction of lithium during the long lifetimes of these old stars is the leading explanation, but the issue remains open and continues to motivate refinements of both stellar and nuclear-reaction modeling.

Summary

Big Bang nucleosynthesis is the universe’s oldest measurement of its own neutrino sector. The primordial abundances of helium and deuterium encode the expansion rate at the moment when weak reactions froze out, and so the effective number of relativistic neutrino species at that epoch. The current BBN constraint, $N_{\mathrm{eff}}^{\mathrm{BBN}}=2.86\pm 0.15$, is fully consistent with the Standard-Model expectation, and combined with the independent CMB constraint at recombination it tightly limits any beyond-Standard-Model physics that would affect the radiation content of the universe between the first three minutes and several hundred thousand years later. The lithium problem remains the one significant tension in an otherwise remarkable agreement between observed light-element abundances and theoretical predictions.