Neutrinos in the Early Universe

Neutrinos were in full thermal equilibrium with photons, electrons, and positrons during the first second after the Big Bang. As the universe expanded and cooled, their weak-interaction rates dropped below the expansion rate, and they decoupled — a transition that set both the present cosmic neutrino density and the stage for primordial nucleosynthesis.

Decoupling

The weak-interaction rate scales as $\Gamma \propto G_F^2{T}^5$; the Hubble expansion rate at high temperature scales as $H\propto T^2/M_{\text{Pl}}$. Equating the two gives the decoupling temperature $T_{\text{dec}}\sim 1\text{ MeV}$ at about $t\sim 1$ s. Neutrino-electron scattering kept ${\nu }_e$ coupled slightly longer than ${\nu }_{\mu }$ and ${\nu }_{\tau }$, giving the electron-flavor distribution a small deviation from perfect Fermi-Dirac. This tiny asymmetry affects primordial helium at the 0.01% level.

After decoupling, the neutrino population red-shifts but no longer thermally exchanges energy with photons. When electrons and positrons subsequently annihilate, the photon bath is heated but the neutrinos are not — hence the familiar temperature ratio $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$.

Big Bang Nucleosynthesis (BBN)

Around three minutes after the Big Bang, the universe had cooled enough for light nuclei to form. The neutron-to-proton ratio at freeze-out set the primordial composition, and this ratio was directly set by competing weak reactions with neutrinos: ${\nu }_e+n\leftrightarrow p+e^{-},{\bar{\nu }}_e+p\leftrightarrow n+e^{+}$ The freeze-out ratio $n/p\approx 1/6$ depends on the neutrino temperature and the expansion rate at that epoch.

Two cosmological parameters are most tightly constrained by BBN:

• Baryon-to-photon ratio $\eta$, cross-checked against CMB measurements
• Number of neutrino species $N_{\text{eff}}$, through its effect on the expansion rate

BBN gives $N_{\text{eff}}^{\text{BBN}}\approx 2.88\pm 0.27$, consistent with CMB-derived values and with the Standard Model value of 3.044. This consistency across completely different epochs is one of the most remarkable successes of modern cosmology.

Neutrinos and the baryon asymmetry

The universe contains far more matter than antimatter. The observed baryon-to-photon ratio is $\eta \sim 6\times 10^{-10}$ — small but significantly non-zero. Sakharov’s three conditions for generating this asymmetry require:

1. Baryon-number violation
2. C and CP violation
3. Departure from thermal equilibrium

In the leptogenesis scenario, these conditions are first satisfied in the lepton sector, through the decay of heavy right-handed neutrinos with CP-violating Yukawa couplings. The resulting lepton asymmetry is then partially converted to a baryon asymmetry through Standard Model sphaleron processes. Leptogenesis works most naturally if the light neutrinos are Majorana — giving additional phenomenological motivation for neutrinoless double beta decay searches.

Neutrinos and structure formation

Because neutrinos are relativistic at decoupling, their thermal velocities imprint a free-streaming length below which they cannot cluster. For $\sum m_{\nu }$ of order 0.1 eV, this scale today is ~100 Mpc. Below that scale, the matter power spectrum is suppressed by roughly $\Delta P/P\approx -8f_{\nu }$ where $f_{\nu }={\Omega }_{\nu }/{\Omega }_m$.

Cosmological neutrino-mass bounds derive from this suppression. High-$k$ galaxy clustering data (from BOSS, eBOSS, DESI) combined with CMB measurements (from Planck, and eventually CMB-S4 and Simons Observatory) are currently approaching sensitivity to the minimum $\sum m_{\nu }$ allowed by oscillation data.

Timeline summary

Neutrinos are witnesses to the first second of cosmic history. Every observational bound on their properties is simultaneously a statement about that epoch — and new experimental routes (precision CMB spectral distortions, direct relic detection) are steadily sharpening that picture.