Neutrino decoupling and relic temperature

Initially, active neutrinos, antineutrinos, electrons, positrons, and photons share a common temperature in the standard zero-chemical-potential approximation. In the instantaneous-decoupling model, remove neutrinos from the coupled bath before electron-positron annihilation. Their thermal-shaped distribution then redshifts with a temperature parameter proportional to inverse scale factor. The electromagnetic sector retains the entropy of photons and electron-positron pairs, so its temperature follows a different cooling law across annihilation.

The electromagnetic entropy count changes from eleven halves to two. Compare the separately redshifting neutrino temperature with the photon temperature after that change. The cube root follows from entropy, rather than from the fourth-power radiation energy density. Counting three neutrino flavors inside the electromagnetic entropy budget after decoupling would transfer heat to a sector that the approximation explicitly disconnected and would spoil the temperature-ratio derivation.

Define the effective neutrino number through the total nonphoton radiation density relative to the ideal temperature-ratio normalization. Additional relativistic species can contribute to this parameter even if they are not neutrinos. Conversely, spectral distortions can change it without adding a particle. The definition is useful during epochs when the contributing species are ultrarelativistic. Once a neutrino becomes nonrelativistic, its rest-mass energy must be computed from its distribution instead of using the radiation formula unchanged.

Froustey, Pitrou, and Volpe's 2020 calculation predicts $N_{\rm eff}=3.0440$ within its stated standard-cosmology assumptions, including flavor oscillations and finite-temperature electromagnetic corrections. It is a theoretical result, not a measurement of the number of flavors. Weak interactions still transfer some entropy during electron-positron annihilation, and the resulting spectra are slightly nonthermal. A single exact common neutrino temperature cannot summarize every momentum-dependent feature, even though effective temperatures are convenient approximations.

An earlier-decoupling species misses more entropy transfers. Its later temperature relative to neutrinos follows from the entropy count of the bath that remains coupled after it leaves. The count must exclude the already-decoupled species itself and reflect the actual particle model. A high-temperature Standard Model count is therefore a model input, not an experimentally established census of every possible particle at arbitrarily high energies. An intervening exotic decay would modify the relation.

Cosmological observations constrain radiation through its effects on expansion and perturbations; they do not simply place a thermometer in a neutrino bath. Inferring a relic abundance requires assumptions about masses, distributions, initial conditions, and other cosmological parameters. The ideal ratio remains valuable because it exposes the entropy mechanism and supplies a baseline for corrections. Its usefulness should not blur the distinction between a derived approximation, a precision kinetic prediction, and a data-dependent parameter estimate.