When Do Neutrinos Leave Thermal Equilibrium?

Neutrinos permeate the cosmos. These nearly massless fermions were in thermal contact with the primordial plasma in the first few seconds after the Big Bang, exchanging energy through weak interactions with electrons, positrons, and nucleons. As the universe expanded and cooled, the interaction rate fell below the Hubble expansion rate, causing neutrinos to decouple and stream freely through space. This event, called neutrino freeze-out, set the relic temperature and density of the cosmic neutrino background and influenced primordial nucleosynthesis. Calculating the decoupling temperature provides insight into early-universe dynamics and the conditions under which the weak force ceased to keep neutrinos in equilibrium with other particles.

Our calculator is built upon the classic freeze-out criterion that equates the interaction rate $\Gamma$ with the Hubble expansion rate $H$. In the radiation-dominated era, the expansion rate is given by $H=1.66\sqrt{g{*}_{}}\frac{T^2}{M}$, where $g{*}_{}$ is the effective number of relativistic degrees of freedom, $T$ is the temperature in energy units, and $M{\mathrm{Pl}}_{}$ is the Planck mass. The interaction rate for neutrinos is dominated by weak processes such as $\nu e+e{}^{\pm }\to \nu e+e{}^{\pm }$ and can be approximated by $\Gamma \approx G^{F_{}}{T}^5$, where $G{F}_{}$ is the Fermi constant that characterizes the strength of the weak force.

Setting $\Gamma =H$ and solving for $T$ yields the decoupling temperature: $T\_\mathrm{dec}={\frac{1.66}{G^{F_{}}}\sqrt{g{*}_{}}}^{\frac{1}{3}}$. Inserting numerical constants, this evaluates to about 1 MeV for $g{*}_{}=10.75$, typical for the epoch just before electron–positron annihilation. Our tool implements this relation directly, letting users adjust $g{*}_{}$ or $G{F}_{}$ to explore beyond-standard-model scenarios or different thermal histories.

The freeze-out temperature in turn determines the neutrino decoupling redshift and cosmic time. The redshift is related to the temperature through the scaling $1+z=\frac{T}{T}$, where $T{0}_{}$ is the present cosmic microwave background temperature. Because neutrinos decouple before electron–positron annihilation, their temperature today is slightly lower than that of photons by a factor of $\text{exception 25:}$, an effect discussed in more detail below. The cosmic time corresponding to a given temperature in the radiation era can be approximated by $t\approx 0.301\frac{M}{{\mathrm{Pl}}_{}}$ when natural units are converted to seconds using $\hslash$. Our calculator outputs both the redshift and the corresponding time in seconds, letting users contextualize neutrino decoupling within the early chronology of the universe.

To appreciate the physics behind the freeze-out condition, consider the scaling of the relevant rates with temperature. The weak interaction rate scales as $T^5$ because of the phase-space density and matrix element behavior of relativistic scattering in a thermal bath. The expansion rate scales as $T^2$ in the radiation era, reflecting how the energy density of relativistic particles drives the cosmic expansion. At high temperatures, the interaction rate wins and neutrinos remain coupled; at low temperatures, the expansion rate dominates and interactions become too infrequent. The power-law forms imply that the ratio $\frac{\Gamma }{H}$ scales as $T^3$, so once the balance tips toward expansion, decoupling proceeds rapidly.

In the standard cosmological model, neutrino decoupling occurs at $T\approx 1\mathrm{MeV}$, corresponding to a cosmic time of about one second and a redshift around $1010$. Soon after, electrons and positrons annihilate, heating the photons relative to the decoupled neutrinos. Entropy conservation implies that the neutrino temperature today is $\frac{4}{11}\text{exception 25:}$ times the photon temperature, resulting in a cosmic neutrino background at about 1.95 K. The neutrino number density is then $n\approx 112\mathrm{cm}-3$ per flavor if no lepton asymmetry is present. While direct detection of this background remains a formidable challenge, its existence is a robust prediction of Big Bang cosmology.

Exploring variations in $g{*}_{}$ or $G{F}_{}$ offers insight into physics beyond the standard model. Additional relativistic species, often parameterized as extra "dark radiation," would raise $g{*}_{}$, increasing the expansion rate and causing earlier decoupling at higher temperatures. Conversely, a weaker effective Fermi constant would delay decoupling. Such modifications can leave imprints in the primordial element abundances and the cosmic microwave background anisotropies, making precise knowledge of neutrino freeze-out crucial for cosmological parameter inference. Our calculator can be used to rapidly scan these scenarios, complementing more detailed numerical treatments.

The table below illustrates how the decoupling temperature and redshift vary with different choices of $g{*}_{}$. For simplicity we keep $G{F}_{}$ fixed at its measured value, highlighting the sensitivity to the relativistic content of the universe.

g*	Tdec (MeV)	zdec
5	0.73	3.1×109
10.75	1.00	4.3×1010
20	1.26	5.4×1010

While the temperature changes only modestly across this range, the redshift span reflects how rapidly the universe expands during radiation domination. Small variations in $g{*}_{}$ at these early times can have outsized consequences for later epochs, such as the synthesis of light elements. Accurate estimates of the decoupling temperature thus provide essential boundary conditions for numerical codes that model Big Bang nucleosynthesis.

Beyond cosmology, understanding neutrino freeze-out informs particle physics. For example, constraints on sterile neutrino properties often involve their contribution to $g{*}_{}$ and the impact on decoupling. Scenarios with nonstandard interactions, such as neutrino magnetic moments or secret gauge forces, can alter the effective interaction rate $\Gamma$, leading to later decoupling and potentially observable distortions in the cosmic neutrino spectrum. Our calculator, while simple, gives a transparent first approximation for such studies.

When using the tool, enter the desired $g{*}_{}$ and $G{F}_{}$. The script computes $T$_dec in MeV, the associated redshift, and the cosmic time in seconds. It also categorizes the result: temperatures below 0.5 MeV are labeled "late decoupling"; those between 0.5 and 2 MeV are "standard," and higher values are "early decoupling." Because the calculation is performed entirely in your browser, no data are sent elsewhere, making the tool suitable for quick exploratory analysis or classroom demonstrations.

In summary, the neutrino decoupling temperature encapsulates the moment when the weak force relinquished control over the neutrino sector, allowing these particles to free-stream and form a cosmic relic background. This event occurred roughly one second after the Big Bang, at temperatures around a mega-electron-volt, and set the stage for subsequent processes such as nucleosynthesis and photon decoupling. By providing an easy way to estimate the freeze-out temperature and its associated cosmic parameters, this calculator helps illuminate one of the early universe's pivotal transitions.