Neutrino Decoupling and Relic Radiation

The cosmic neutrino background is a relic of weak-interaction decoupling near the MeV era. Its simplest prediction is not merely that neutrinos survive, but that their temperature becomes lower than the photon temperature when electron–positron pairs transfer entropy to the still-coupled electromagnetic bath. The ideal ratio $(4/11)^{1/3}$ follows from separate entropy conservation; precision decoupling replaces the ideal $N_{\rm eff}=3$ by $N_{\rm eff}=3.044$ and leaves small nonthermal spectral distortions.

Required background. States, Processes, and the Laws of Thermodynamics supplies Thermodynamics and Statistical Mechanics for equilibrium state variables, entropy, temperature, and the first law. Statistical Ensembles and Partition Functions and Quantum Statistical Mechanics and Density Operators develop the equilibrium distributions used below, while Thermodynamics in an Expanding Universe fixes the bath, effective-degree-of-freedom, and separate-sector entropy conventions.

Helpful background. The Einstein Equation supplies the Einstein-equation convention, while The Friedmann Equations supplies the FLRW expansion rate; Reaction Rates, Decoupling, and Freeze-Out develops the general $\Gamma/H$ criterion; Relativistic Matter and Compact Objects supplies Relativistic Matter and Compact Objects thermodynamic and stress-energy context; and Classical Field Modes and Canonical Variables supplies the Flat-Spacetime Quantum Field Theory mode convention. Free Fields, Creation Operators, and Fock Space provides occupation-state counting.

Weak decoupling near the MeV scale

Use $\hbar=c=k_{\rm B}=1$. Let $t$ be cosmic time, $H=\dot a/a$, and $T_\gamma$ the temperature of the photon–plasma bath. The Fermi constant is denoted by $G_F$. At $T_\gamma\ll m_W,m_Z$, a relativistic target density scales as $n\sim T_\gamma^3$, and four-Fermi dimensional analysis gives

$$\langle\sigma v\rangle_{\rm weak} \sim G_F^2T_\gamma^2, \qquad \Gamma_\nu\sim n\langle\sigma v\rangle \sim G_F^2T_\gamma^5.$$

During radiation domination,

$$H=1.66\,g_*^{1/2}(T_\gamma) \frac{T_\gamma^2}{M_{\rm Pl}}, \qquad M_{\rm Pl}=G^{-1/2},$$

where $g_*$ is defined relative to $T_\gamma$ and $M_{\rm Pl}$ is the nonreduced Planck mass. Thus

$$\frac{\Gamma_\nu}{H} \sim \frac{G_F^2M_{\rm Pl}}{1.66\,g_*^{1/2}}T_\gamma^3.$$

The simple $T_\gamma^3$ scaling holds when $g_*$ varies slowly. Taking $g_*=10.75$ and the omitted reaction coefficient to be unity gives $T_{\rm dec}\sim1.5\,\mathrm{MeV}$, an order-of-magnitude check rather than a precision prediction. Electron and muon/tau flavors have different charged-current channels, and the collision term decays gradually rather than switching off. The rate estimate and ideal decoupling limit are developed in Olive and Peacock 2025/2026, §22.3.3, printed pp. 12–13, eqs. (22.45)–(22.51) (Official PDF).

Let $p$ be physical momentum magnitude and $q=ap$ the corresponding comoving momentum. Once collisions and coherent flavor conversion are negligible, each incoherent mass-eigenstate occupation in a homogeneous, isotropic universe obeys

$$\left(\partial_t-Hp\partial_p\right)f_\nu=0.$$

The comoving momentum is constant along a trajectory, so $f_\nu(q)$ is conserved. An initially thermal, zero-chemical-potential relativistic spectrum can still be parameterized by $T_\nu\propto a^{-1}$, even though it is no longer in thermal contact with the plasma. Here $T_\nu$ is the momentum scale of the frozen distribution, not a general thermodynamic equilibrium temperature. When a mass eigenstate later becomes nonrelativistic, its momentum continues to redshift without restoring kinetic equilibrium.

Separate entropy conservation and the 4/11 ratio

For the ideal estimate, assume three thermally populated active neutrino–antineutrino helicity pairs and negligible electron and neutrino chemical potentials. Choose an initial endpoint $a_-$ after idealized neutrino decoupling but while $e^\pm$ are still ultrarelativistic, and a final endpoint $a_+$ after the pairs have disappeared. At $a_-$, photons and $e^\pm$ still share electromagnetic entropy, with

$$g_{*s,\gamma e}^{\rm before} =2+\frac78(2+2)=\frac{11}{2}.$$

At $a_+$ only the two photon polarizations remain, so $g_{*s,\gamma}^{\rm after}=2$. In this ideal separation of events, the decoupled neutrinos and electromagnetic bath still have a common temperature $T_-$ at $a_-$ because no intervening entropy transfer has occurred. Separate entropy conservation and collisionless redshifting then give

$$\frac{11}{2}T_-^3a_-^3 =2T_{\gamma,+}^3a_+^3, \qquad T_{\nu,+}=T_-\frac{a_-}{a_+}.$$

Eliminating the endpoint scale factors yields

$$\boxed{ \frac{T_\nu}{T_\gamma} =\left( \frac{g_{*s,\gamma}^{\rm after}} {g_{*s,\gamma e}^{\rm before}} \right)^{1/3} =\left(\frac{4}{11}\right)^{1/3} }.$$

This check depends on the sector boundary. The total $S_{\gamma e}+S_\nu$ is conserved in the ideal treatment, but the two terms must be evolved separately once neutrinos decouple. Assigning photons, electrons, and neutrinos one common temperature and a single $g_{*s}$ would incorrectly erase the relative reheating. Conversely, treating annihilation as creation of new total entropy would miss that the ideal result is an internal redistribution within an adiabatic electromagnetic sector.

For three ultrarelativistic active species, with one neutrino and one antineutrino helicity state per species, the ideal density after annihilation is

$$\rho_\nu =3\left(\frac78\right)2\frac{\pi^2}{30}T_\nu^4.$$

Since $\rho_\gamma=2(\pi^2/30)T_\gamma^4$, define the conventional reference momentum temperature $T_\nu^{(0)}=(4/11)^{1/3}T_\gamma$. The standard radiation-density parameterization is

$$\boxed{ \rho_{\rm rad} =\rho_\gamma\left[ 1+\frac78\left(\frac{4}{11}\right)^{4/3}N_{\rm eff} \right] }.$$

In the instantaneous, zero-chemical-potential limit $N_{\rm eff}=3$. This normalization is Lesgourgues and Verde 2025/2026, §26.1, printed p. 3, eq. (26.1) (Official PDF).

Why precision decoupling gives 3.044

Real neutrino decoupling overlaps the beginning of $e^\pm$ annihilation. Residual weak interactions transfer a small share of the entropy to neutrinos and produce flavor-dependent distortions rather than an exact Fermi–Dirac spectrum. Neutrino oscillations redistribute those distortions, and finite-temperature quantum-electrodynamic corrections change the plasma equation of state. A precision treatment evolves a flavor density matrix with collision and oscillation terms rather than the scalar collisionless equation used after freeze-out. Modern calculations package the resulting relativistic energy density as

$$\boxed{N_{\rm eff}^{\rm SM}=3.044}.$$

Froustey, Pitrou, and Volpe derive the quantum kinetic equations and full collision term in Froustey, Pitrou, and Volpe 2020, §2.2.3, printed pp. 8–9, eqs. (2.20)–(2.25) (Open PDF), and report the distorted spectra, QED equation-of-state correction, and $N_{\rm eff}=3.04397\simeq3.0440$ in Froustey, Pitrou, and Volpe 2020, §4.3, printed pp. 16–18, eqs. (4.8)–(4.10), figs. 3–4, and table 1 (Open PDF). The current review value and its assumptions are summarized in Lesgourgues and Verde 2025/2026, §26.1, printed p. 3, eq. (26.1) and the following paragraphs (Official PDF).

$N_{\rm eff}$ is an energy-density convention, not a literal count of particles. The conventional factor $(4/11)^{4/3}$ is retained in its definition, while noninstantaneous reheating and spectral distortions are absorbed into $N_{\rm eff}$. Standard background bookkeeping therefore keeps $T_\nu^{(0)}=(4/11)^{1/3}T_\gamma$ and uses $3.044$ in the density formula; momentum-dependent precision calculations instead use the actual distorted distributions. Raising a thermal $T_\nu$ to represent the same reheating and also inserting $3.044$ would double count unless the convention is explicitly remapped. Additional light relics, nonstandard neutrino interactions, or entropy transfer can change the same parameter.

From radiation to free-streaming matter

While relativistic, neutrinos contribute to $H$ through $\rho_{\rm rad}$ and therefore shift radiation–matter equality and characteristic acoustic and diffusion scales. Their collisionless perturbations also develop anisotropic stress. That free-streaming stress changes the phase and amplitude of acoustic oscillations in a way that an equally dense tightly coupled relativistic fluid does not. Background density and perturbation transport are thus separate parts of the physical model. These background and acoustic effects are reviewed in Lesgourgues and Verde 2025/2026, §§26.2.1–26.2.2, printed pp. 4–5 (Official PDF).

For an ideal zero-chemical-potential Fermi–Dirac momentum distribution,

$$\frac{\langle p\rangle}{T_\nu^{(0)}} =\frac{7\pi^4}{180\,\zeta(3)} \simeq3.151.$$

Adopt $a_0=1$, so $1+z=a^{-1}$, and denote the present reference momentum temperature by $T_{\nu0}^{(0)}=(4/11)^{1/3}T_{\gamma0}$. A mass eigenstate makes a gradual relativistic-to-nonrelativistic transition near the ideal estimate

$$1+z_{\rm nr} \simeq\frac{m_i}{3.151\,T_{\nu0}^{(0)}}.$$

The actual transition is broad, and precision distortions perturb this one-number estimate. Afterward the eigenstate contributes a hot matter component but retains a large velocity dispersion. It clusters inefficiently on wavelengths shorter than its free-streaming length, equivalently above the corresponding free-streaming wavenumber, and alters the growth of structure. The momentum-redshifting and transition estimate appear in Lesgourgues and Verde 2025/2026, §26.1, printed pp. 3–4 (Official PDF), and the massive-neutrino CMB and structure effects in Lesgourgues and Verde 2025/2026, §§26.2.3–26.2.4, printed pp. 5–7 (Official PDF). The detailed evolution belongs to Einstein–Boltzmann Evolution and Linear Growth of Matter Perturbations; the bridge here is that the same frozen $f_\nu(q)$ determines both the early radiation density and the later velocity distribution.

Common pitfalls

Calling $N_{\rm eff}$ the number of neutrino flavors. It measures radiation density in a specified unit. Three active flavors give 3.044 in the Standard Model, and other radiation can contribute to the same quantity.

Heating a sector after it has decoupled. Entropy released by $e^\pm$ annihilation reheats the coupled electromagnetic bath. In the ideal limit it does not reheat neutrinos; precision corrections arise only because decoupling is incomplete.

Equating free streaming with no gravitational effect. Collisionless neutrinos do not cluster like cold matter on every scale, but their background density, pressure, momentum flux, and anisotropic stress still gravitate.

Exercises

A real, massless scalar $X$ with negligible chemical potential has $T_X=\xi T_\nu^{(0)}$ and remains relativistic. Compute its contribution $\Delta N_{\rm eff}$ in the standard convention. Repeat for a massless Weyl fermion and antifermion with one populated helicity each—two states in total—at the same temperature. How would the answer change for a Dirac pair with four populated states in total?

Solution

The scalar energy density is

$$\rho_X=\frac{\pi^2}{30}T_X^4.$$

One unit of $N_{\rm eff}$ uses the conventional reference density of one neutrino–antineutrino pair,

$$\rho_{\nu,1} =\frac78\,2\frac{\pi^2}{30}\left(T_\nu^{(0)}\right)^4.$$

Therefore

$$\boxed{ \Delta N_{\rm eff}^{\rm scalar} =\frac{\rho_X}{\rho_{\nu,1}} =\frac47\xi^4 }.$$

For the two-state Weyl pair, the factor $(7/8)2$ matches the reference neutrino unit, so

$$\boxed{ \Delta N_{\rm eff}^{\rm fermion}=\xi^4 }.$$

A four-state Dirac pair has twice that density and therefore contributes $\Delta N_{\rm eff}^{\rm Dirac}=2\xi^4$.

The fourth power shows why entropy dilution, which fixes the temperature ratio at decoupling, strongly controls the later radiation density.

References

• Froustey, Julien, Cyril Pitrou, and Maria Cristina Volpe. “Neutrino Decoupling Including Flavour Oscillations and Primordial Nucleosynthesis.” Journal of Cosmology and Astroparticle Physics 2020 (12): 015. doi:10.1088/1475-7516/2020/12/015; arXiv:2008.01074v2 Open PDF.
• Lesgourgues, Julien, and Licia Verde. “Neutrinos in Cosmology.” In F. Takahashi et al. (Particle Data Group), Review of Particle Physics, International Journal of Modern Physics A 41, 2630011 (2026). Review revised August 2025; issue dated 1 June 2026. Official PDF.
• Olive, K. A., and J. A. Peacock. “Big-Bang Cosmology.” In F. Takahashi et al. (Particle Data Group), Review of Particle Physics, International Journal of Modern Physics A 41, 2630011 (2026). Review revised August 2025; issue dated 1 June 2026. Official PDF.