Modern cosmological theories imply that the universe is filled with a shallow degenerate Fermi sea of neutrinos. In the steady state and oscillating models (and perhaps also the "big bang" theories) it can be shown rigorously that the proportion of filled neutrino levels (plus the proportion of filled antineutrino levels) is precisely one up to a finite Fermi energy ${E}_{F}$. The proof takes into account both absorption and the repressive effects of already filled levels on neutrino emission. Experiment shows that ${E}_{F}\ensuremath{\le}200$ eV for antineutrinos and ${E}_{F}\ensuremath{\le}1000$ eV for neutrinos. The degenerate neutrinos could be observed (if ${E}_{F}g10$ eV) by looking for apparent violations of energy conservation in ${\ensuremath{\beta}}^{\ensuremath{-}}$ decay. In the steady state and evolutionary cosmologies ${E}_{F}$ is much too low to ever be observed, but in the oscillating cosmologies ${E}_{F}\ensuremath{\simeq}5{R}_{c}$ MeV, where ${R}_{c}$ is the minimum radius of the universe in units of its present radius; thus experiment already shows that the universe will contract by a factor over ${10}^{3}$, if at all. Astronomical evidence plus Einstein's field equation (without cosmological constant) require in an oscillating cosmology that ${E}_{F}l2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ eV (so ${R}_{c}l{10}^{\ensuremath{-}9}$) and suggest that higher energy neutrinos may represent the bulk of the energy of the universe. A model universe incorporating this idea is constructed.