Question

Just as the Universe has a cosmic microwave background dating back to the time when the Universe was opaque to photons, it has a cosmic neutrino background dating back to the earlier time when the Universe was opaque to neutrinos. The calculated number density of cosmic neutrinos is n_ν = 3.36 × 10⁸ m^-3. (a) How many cosmic neutrinos are inside (and streaming through) the Earth at any given moment? (b) What average neutrino mass, mν, would be required to make the mass density of the cosmic neutrino equal to the critical density of the Universe?

Answer 1

a] Number density of cosmic neutrinos = n_v = 3.36 x 10⁸/m³

Volume of Earth =

where R = Radius of the Earth = 6378.1 km = 6378100 m

so, number of cosmic neutrinos which are inside Earth at any given moment is:

b] Critical density of the universe is the density at which the universe is stable. It is approximately, n_o = 10^-26 kg/m³

mass density of cosmic neutrino = mass x number density = m [3.36 x 10⁸] kg/m³

therefore, average neutrino mass mv = n_o / [3.36 x 10⁸] = 2.976 x 10^-35 kg.