Neutrino Temperature

Neutrino Temperature

The universe is filled not just with the Cosmic Microwave Background (CMB) photons, but also with a Cosmic Neutrino Background (CνB). These relic neutrinos decoupled from the thermal bath when the universe was about 1 second old.

Neutrino Decoupling

Neutrinos decoupled at $T \approx 2$–$3$ MeV, just before electron-positron annihilation. When $e^+ e^-$ pairs annihilated (at $T \approx 0.5$ MeV), they dumped entropy into the photon bath — but not into the already-decoupled neutrinos.

By conservation of entropy in the photon-electron plasma:

$\frac{T_\nu}{T_\gamma} = \left(\frac{4}{11}\right)^{1/3} \approx 0.7138$

Neutrino Temperature Today

The CMB temperature today is $T_\gamma = 2.725$ K, so the cosmic neutrino background temperature is:

$T_\nu = T_\gamma \left(\frac{4}{11}\right)^{1/3} \approx 1.945 \text{ K}$

These neutrinos have never been directly detected (they are extremely weakly interacting), but their gravitational effects are well-established.

Effective Relativistic Degrees of Freedom

The total radiation energy density is written in terms of an effective number of degrees of freedom $g_{\rm eff}$:

$g_{\rm eff} = 2 + \frac{7}{8} \cdot 2 \cdot N_{\rm eff} \cdot \left(\frac{T_\nu}{T_\gamma}\right)^4 = 2 + \frac{7}{8} \cdot 2 \cdot N_{\rm eff} \cdot \left(\frac{4}{11}\right)^{4/3}$

The factor 2 comes from photon polarisations. Each neutrino family contributes $\frac{7}{8} \times 2$ (fermion factor × particle + antiparticle). For the standard 3 neutrino families: $g_{\rm eff} \approx 3.36$.

Your Task

Implement three functions. All constants must be defined inside each function.

• neutrino_to_photon_temp_ratio() — returns $(4/11)^{1/3}$
• neutrino_temperature_K(T_cmb_K) — returns $T_\nu = T_\gamma \cdot (4/11)^{1/3}$
• effective_relativistic_dof(N_eff) — returns $g_{\rm eff} = 2 + \frac{7}{8} \cdot 2 \cdot N_{\rm eff} \cdot (4/11)^{4/3}$