Neutrino Decoupling Temperature Calculator

Formula Used

Neutrinos decouple when their interaction rate drops below the cosmic expansion rate.

• Weak interaction rate: Γ(T) ≈ CΓ · GF2 · T5
• Radiation-era Hubble rate: H(T) ≈ 1.66 · √g* · T2 / MPl
• Decoupling temperature: Tdec ≈ [1.66 √g* / (CΓ GF2 MPl)]1/3

This is a standard order-of-magnitude freeze-out estimate. Detailed treatments include channel-specific constants and finite-temperature effects, which are absorbed here into CΓ.

How to Use This Calculator

1. Set g* for the temperature range you care about.
2. Keep Hubble prefactor at 1.66 for standard expansion.
3. Use CΓ to model rate prefactors and channel details.
4. Select a Planck mass option matching your convention.
5. Optionally enter uncertainties to estimate a plausible temperature band.
6. Press Calculate, then export via CSV or PDF buttons.

Example Data Table

The “Estimated” column shows qualitative expectations; your computed values appear in the results panel after calculation.

Neutrino Decoupling Temperature in Practice

1) Why neutrino decoupling matters

In the early universe, neutrinos were kept in thermal contact with the plasma by weak interactions. As the universe expanded and cooled, these reactions became too slow to maintain equilibrium. The temperature where neutrinos effectively fall out of equilibrium helps set the neutrino temperature relative to the photon bath and influences light‑element yields, radiation density, and timing benchmarks in the first seconds.

2) Interaction rate scaling

For MeV‑scale energies, weak rates can be summarized by a power law, Γ(T) ≈ CΓ G_F^2 T^5. The fifth‑power dependence is crucial: even a modest cooling quickly suppresses reaction rates. The factor CΓ is an adjustable, order‑one knob that absorbs channel counting, spin sums, and simplified thermal averaging so you can explore “faster” or “slower” effective coupling scenarios.

3) Expansion rate and g*

During radiation domination, the Hubble rate scales as H(T) ∝ √g* · T^2. The parameter g* counts effective relativistic degrees of freedom contributing to the energy density. Around the MeV epoch, a commonly used benchmark is g* ≈ 10.75, reflecting photons, electron‑positron pairs, and three neutrino species treated as relativistic.

4) Solving for the decoupling temperature

Freeze‑out is estimated by the balance condition Γ(T) = H(T). With the forms above, the solution becomes a compact cube‑root relation: T_dec ≈ [1.66 √g* / (CΓ G_F^2 M_Pl)]^(1/3). This calculator implements that expression directly and reports results in MeV, GeV, and kelvin for convenient cross‑checks.

5) Typical MeV‑scale benchmarks

Using standard inputs (g* = 10.75, CΓ = 1, and the usual radiation prefactor), the estimate lands at a few MeV. This aligns with the physical picture that neutrinos stop tracking the electromagnetic plasma shortly before or around electron‑positron annihilation, after which photons are heated relative to neutrinos.

6) Sensitivity to your assumptions

The cube‑root dependence makes the result relatively insensitive to uncertain prefactors. Doubling CΓ lowers T_dec by only about 2^(−1/3) (~20%). Increasing g* raises T_dec as (g*)^{1/6}, so even large changes in particle content shift the decoupling temperature gradually.

7) Interpreting MeV, GeV, and kelvin

Thermal energies are often quoted in MeV in early‑universe work, while kelvin is useful for connecting to broader thermodynamics. The calculator converts using 1 GeV ≈ 1.16045×10^13 K. Seeing multiple units side‑by‑side helps validate that the order of magnitude is consistent before you export a report.

8) Limits of the simple estimate

This tool is designed for controlled exploration, not a full Boltzmann‑transport solution. Realistic decoupling is gradual, depends on energy‑dependent scattering, and is affected by finite‑temperature QED and neutrino flavor physics. Treat CΓ and optional uncertainty bands as a way to bracket outcomes rather than a substitute for precision calculations.

FAQs

1) What does “decoupling” mean here?

It means weak reactions become slower than cosmic expansion, so neutrinos stop maintaining thermal equilibrium with the plasma. The calculator estimates the temperature where Γ(T) = H(T).

2) Why is the answer typically a few MeV?

At MeV energies, weak rates drop rapidly as T^5 while expansion falls only as T^2. Their crossover naturally occurs near MeV scales for standard cosmological parameters.

3) What should I use for g*?

Use the effective relativistic degrees of freedom appropriate to the epoch you model. A common benchmark near the MeV era is g* ≈ 10.75, but you can explore alternatives for nonstandard particle content.

4) What does CΓ represent?

CΓ is an order‑one prefactor that summarizes detailed scattering channels and thermal averages. If you believe rates are effectively stronger, increase CΓ; if weaker, decrease it.

5) Why does changing parameters move the result slowly?

The solution is a cube‑root: T_dec ∝ (√g*/CΓ)^{1/3}. Cube‑roots damp changes, so even large input variations typically shift the temperature by tens of percent, not orders of magnitude.

6) Which Planck mass option should I pick?

Choose the convention used in your reference. Some formulas are written with the Planck mass, others with the reduced Planck mass. The calculator lets you switch so your H(T) convention stays consistent.

7) How should I interpret the uncertainty band?

It is a simple extrema estimate from your ±g* and ±CΓ inputs, not a statistical error propagation. Use it to bracket plausible outcomes when your model inputs are uncertain or scenario‑dependent.