Astroparticle Physics

13.2 Motivation for Dark Matter 273equal numbers of blackbody neutrinos and blackbody photons.With N ≈ 300 neutrinos/cm 3 and Ω = 1 (correspondingto the critical density of ϱ c ≈ 1 × 10 −29 g/cm 3 at an ageof the universe of approximately 1.4 × 10 10 years) one obtains4N ∑ ∑m ν ≤ ϱ c , mν ≤ 20 eV . (13.12)The sum extends over all sequential neutrinos includingtheir antiparticles. For the three known neutrino generationsone has ∑ m ν = 2(m νe + m νµ + m ντ ). The consequenceof (13.12) is that for each individual neutrino flavour a masslimit can be derived:m ν ≤ 10 eV . (13.13)It is interesting to see that on the basis of these simplecosmological arguments the mass limit for the τ neutrino asobtained from accelerator experiments can be improved byabout 6 orders of magnitude.If the contribution of neutrino masses to dark matter isassumedtobeΩ ν > 0.1, a similar argument as before providesalso a lower limit for neutrino masses. Under the assumptionof Ω ν > 0.1 (13.12) yields for the sum of massesof all neutrino flavours ∑ m ν > 2 eV. If one assumes thatin the neutrino sector the same mass hierarchy holds as withcharged leptons (m e ≪ m µ ≪ m τ → m νe ≪ m νµ ≪ m ντ ),the mass of the τ neutrino can be limited to the rangeupper summative neutrinomass limit from flat universeupper individual neutrinomass limits from flat universelower neutrino mass limitfrom minimumenergy density1eV≤ m ντ ≤ 10 eV . (13.14)This conclusion, however, rests on the assumption ofΩ ν > 0.1. Recent estimates of Ω ν indicate much smallervalues (Ω ν < 1.5%) and hence the ν τ mass can be substantiallysmaller. Neutrinos with low masses are relativistic andwould constitute in the early universe in thermal equilibriumto the so-called ‘hot dark matter’. With hot dark matter, however,it is extremely difficult to understand the structures inthe universe on small scales (size of galaxies). Therefore,neutrinos are not considered a good candidate for dark matter.There is a possibility to further constrain the margin forneutrino masses. To contribute directly to the dark matterof a galaxy, neutrinos must be gravitationally bound to the4 For simplification c = 1 has been generally used. If numbers,however, have to be worked out, the correct numerical value forthe velocity of light, c ≈ 3 × 10 8 m/s, must be used.hot dark mattergravitational bindingto a galaxy