Astroparticle Physics

274 13 Dark Matterneutrinosas free relativistic fermion gasneutrino mass densitylower neutrino mass limitfrom galaxy dynamicsgalaxy, i.e., their velocity must be smaller than the escapevelocity v f . This allows to calculate a limit for the maximummomentum p max = m ν v f . If the neutrinos in a galaxy aretreated as a free relativistic fermion gas in the lowest energystate (at T = 0), one can derive from the Fermi energyE F = ¯hc(3π 2 n max ) 1/3 = p max c (13.15)an estimation for the neutrino mass density (n max – numberdensity):n max m ν = m4 ν v3 f3π 2 ¯h 3 . (13.16)Since n max m ν must be at least on the order of magnitude ofthe typical density of dark matter in a galaxy, if one wantsto explain its dynamics with neutrino masses, these argumentslead to a lower limit for the neutrino mass. Usingv f = √ 2GM/r, whereM and r are galactic mass and radius,one obtains under plausible assumptions about the neutrinomass density and the size and structure of the galaxym ν > 1eV. (13.17)Again, this argument is based on the assumption that neutrinomasses might contribute substantially to the matterdensity of the universe. These cosmological arguments leaveonly a relatively narrow window for neutrino masses.These considerations are not necessarily in contradictionto the interpretation of results on neutrino oscillations, becausein that case one does not directly measure neutrinomasses but rather the difference of their masses squared.From the deficit of atmospheric muon neutrinos one obtainsδm 2 = m 2 1 − m2 2 = 3 × 10−3 eV 2 . (13.18)If ν µ –ν τ oscillations are responsible for this effect, muonand tau neutrino masses could still be very close withoutneutrino masses getting into conflict with the cosmological mass limits. Onlyfrom oscillations if the known mass hierarchy (m e ≪ m µ ≪ m τ ) fromthe sector of charged leptons is transferred to the neutrinosector and if one further assumes m νµ ≪ m ντ ,theresult(m ντ ≈ 0.05 eV) would be in conflict with the cosmologicallimits, but then the cosmological limits have been derivedunder the assumption that neutrino masses actually play animportant rôle for the matter density in the universe, whichis now known not to be the case.