Neutrino Decoupling Calculator

Neutrino decoupling is a cornerstone of Big Bang cosmology. At temperatures above ~1 MeV, neutrinos were in thermal equilibrium with photons, electrons, and positrons via weak interactions. As the universe expanded and cooled, the weak interaction rate fell below the Hubble expansion rate, causing neutrinos to decouple. This freeze-out sets the relic neutrino background temperature and influences the effective number of relativistic species (N_eff), a key parameter in CMB and BBN analyses.

The decoupling temperature is not sharp but occurs over a range due to finite interaction rates. The Neutrino Decoupling Calculator uses the full Boltzmann equation solution to compute the precise T_dec where Γ_weak ≈ H. This is critical for interpreting Planck CMB data, where N_eff = 2.99 ± 0.17 constrains new physics like sterile neutrinos or dark radiation.

Understanding neutrino decoupling also informs primordial nucleosynthesis (BBN). The neutron-to-proton ratio freezes out near T ~ 0.8 MeV, shortly after neutrino decoupling, making accurate entropy tracking essential for predicting light element abundances like ⁴He and ⁷Li.

Follow these steps to use the Neutrino Decoupling Calculator:

1. CMB Temperature (T₀): Enter the current CMB temperature in mK. Default is 2.725 mK (Planck 2018).
2. N_eff: Enter the effective number of neutrino species. Use 3.046 for Standard Model with QED corrections.
3. Entropy Model: Choose:
   • Standard: Assumes e⁺e⁻ annihilation conserves entropy in photons.
   • Finite Temp: Includes partial neutrino heating (Mangano et al., 2005).
   • Custom: Manually set initial g₀ (advanced users).
4. Click Calculate Decoupling to get:
   • Decoupling temperature T_dec (MeV)
   • Relic neutrino temperature today T_ν (K)
   • Entropy ratio g_*S(T_dec) / g_*S(now)
   • Freeze-out redshift z_dec

Results are based on peer-reviewed papers and include finite-temperature QED corrections when selected. For research, cite: Mangano et al., Nucl. Phys. B 729 (2005).

Use the Neutrino Decoupling Calculator when:

• Teaching Cosmology: Demonstrate how neutrinos decouple and affect N_eff.
• CMB Analysis: Compute expected N_eff for beyond-ΛCDM models.
• BBN Studies: Determine entropy evolution during weak freeze-out.
• Research: Test sterile neutrinos, dark radiation, or modified gravity.
• Public Outreach: Explain why we expect a cosmic neutrino background (CνB).

The relic neutrinos decoupled ~1 second after the Big Bang and now form a background at T_ν ≈ 1.95 K — colder than the CMB due to entropy transfer during e⁺e⁻ annihilation. Detecting this CνB is a frontier goal (PTOLEMY experiment).

The Neutrino Decoupling Calculator is built on rigorous physics:

• Decoupling Condition: Γ_weak(T) = H(T) → T_dec ≈ 1.0 MeV
• Weak Rate: Γ ≈ G_F² T⁵ (Fermi theory)
• Hubble Rate: H ≈ 1.66 g₊¹/² T² / M_Pl
• Entropy Conservation: g₊S T³ = constant (post e⁺e⁻)
• Relic Temperature: T_ν = (4/11)^1/3 T_γ ≈ 1.945 K

After e⁺e⁻ annihilation, photons are reheated, reducing the neutrino-to-photon temperature ratio to (4/11)^1/3. Finite-temperature QED corrections slightly increase N_eff to 3.046.

This calculator uses the full numerical solution from Mangano et al. (2005), accounting for partial neutrino heating and non-instantaneous decoupling. It is the same method used in modern Boltzmann codes like CLASS and CAMB.

For deeper exploration, visit Agri Care Hub for more science tools or read about Neutrino Decoupling on Wikipedia.

The cosmic neutrino background is a fossil of the hot Big Bang, predicted with high precision. While direct detection remains challenging, its imprint on the CMB and BBN is observed. This calculator bridges theory and observation, empowering users to explore one of the universe’s earliest transitions.