Massive neutrinos which are relativistic at the time that...

Massive neutrinos which are relativistic at the time that the weak interactions freeze out will have a number density at the present time equal to that of massless neutrinos, $$ n_{v \bar{v}}=\frac{3}{11} n_\gamma $$ and will contribute a mass density $\rho_{\nu \bar{\nu}}=m_v n_{v \bar{v}}$. If one neutrino family contributes the critical density, compute the mass of this neutrino type. (The masses of the neutrinos are not known, but non-zero values ofless than $1 \mathrm{eV}$ seem to be suggested by present evidence, insufficient to provide the critical density.)

Massive neutrinos which are relativistic at the time that the weak interactions freeze out will have a number density at the present time equal to that of massless neutrinos,
$$
n_{v \bar{v}}=\frac{3}{11} n_\gamma
$$
and will contribute a mass density $\rho_{\nu \bar{\nu}}=m_v n_{v \bar{v}}$. If one neutrino family contributes the critical density, compute the mass of this neutrino type. (The masses of the neutrinos are not known, but non-zero values ofless than $1 \mathrm{eV}$ seem to be suggested by present evidence, insufficient to provide the critical density.)

Key Concepts

Critical Density

Critical density is the threshold energy density that distinguishes whether the universe is open, flat, or closed. It represents the density required for the universe to have zero curvature, and it is a fundamental cosmological parameter that helps determine the overall geometry and fate of the universe.

Neutrino Freeze-Out and Relic Neutrino Number Density

In the early universe, neutrinos decouple from the primordial plasma when the weak interactions freeze out, leaving behind a relic population. This decoupling establishes a fixed number density of neutrinos that, for relativistic neutrinos at freeze-out, scales today as n_? = (3/11) n_?, where n_? is the number density of cosmic microwave background photons.

Neutrino Mass Density Contribution

The mass density contributed by neutrinos is obtained by multiplying the mass of the neutrinos by their relic number density. Equating this product to the critical density allows one to infer the mass that a single neutrino species must have in order to account for all of the critical density, highlighting the relationship between particle physics and cosmological constraints.

Cosmological Constraints on Neutrino Mass

Cosmological observations impose limits on neutrino masses by comparing the computed mass density from relic neutrinos to the critical density of the universe. Even though particle physics experiments suggest neutrino masses in the sub-eV range, these masses are too small to account for the entire critical density, influencing our understanding of dark matter and the overall mass-energy budget of the cosmos.

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